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</div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-分布形式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#分布形式"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>分布形式</span> </div> </a> <ul id="toc-分布形式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-統計力學上的應用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#統計力學上的應用"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>統計力學上的應用</span> </div> </a> <ul id="toc-統計力學上的應用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-數學上的應用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#數學上的應用"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>數學上的應用</span> </div> </a> <ul id="toc-數學上的應用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參見" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參見"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>參見</span> </div> </a> <ul id="toc-參見-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>參考文獻</span> </div> </a> <ul id="toc-參考文獻-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label 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">开关目录</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">玻尔兹曼分布</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="前往另一种语言写成的文章。28种语言可用" > <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-28" 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">28种语言</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%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%A8%D9%88%D9%84%D8%AA%D8%B2%D9%85%D8%A7%D9%86" title="توزيع بولتزمان – 阿拉伯语" lang="ar" hreflang="ar" data-title="توزيع بولتزمان" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_de_Boltzmann" title="Distribució de Boltzmann – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Distribució de Boltzmann" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" 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/Boltzmannovo_rozd%C4%9Blen%C3%AD" title="Boltzmannovo rozdělení – 捷克语" lang="cs" hreflang="cs" data-title="Boltzmannovo rozdělení" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Boltzmann-fordelingen" title="Boltzmann-fordelingen – 丹麦语" lang="da" hreflang="da" data-title="Boltzmann-fordelingen" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Boltzmann-Statistik" title="Boltzmann-Statistik – 德语" lang="de" hreflang="de" data-title="Boltzmann-Statistik" data-language-autonym="Deutsch" data-language-local-name="德语" 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/Boltzmann_distribution" title="Boltzmann distribution – 英语" lang="en" hreflang="en" data-title="Boltzmann distribution" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Boltzmann" title="Distribución de Boltzmann – 西班牙语" lang="es" hreflang="es" data-title="Distribución de Boltzmann" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%D8%A8%D9%88%D9%84%D8%AA%D8%B3%D9%85%D8%A7%D9%86" title="توزیع بولتسمان – 波斯语" lang="fa" hreflang="fa" data-title="توزیع بولتسمان" data-language-autonym="فارسی" data-language-local-name="波斯语" 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/Boltzmannin_jakauma" title="Boltzmannin jakauma – 芬兰语" lang="fi" hreflang="fi" data-title="Boltzmannin jakauma" data-language-autonym="Suomi" data-language-local-name="芬兰语" 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/Distribution_de_Boltzmann" title="Distribution de Boltzmann – 法语" lang="fr" hreflang="fr" data-title="Distribution de Boltzmann" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</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%9C%D7%A6%D7%9E%D7%9F" title="התפלגות בולצמן – 希伯来语" lang="he" hreflang="he" data-title="התפלגות בולצמן" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Boltzmann-eloszl%C3%A1s" title="Boltzmann-eloszlás – 匈牙利语" lang="hu" hreflang="hu" data-title="Boltzmann-eloszlás" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Distribusi_Boltzmann" title="Distribusi Boltzmann – 印度尼西亚语" lang="id" hreflang="id" data-title="Distribusi Boltzmann" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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/Distribuzione_di_Boltzmann" title="Distribuzione di Boltzmann – 意大利语" lang="it" hreflang="it" data-title="Distribuzione di Boltzmann" data-language-autonym="Italiano" data-language-local-name="意大利语" 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%AB%E3%83%84%E3%83%9E%E3%83%B3%E5%88%86%E5%B8%83" title="ボルツマン分布 – 日语" lang="ja" hreflang="ja" data-title="ボルツマン分布" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%BC%EC%B8%A0%EB%A7%8C_%EB%B6%84%ED%8F%AC" title="볼츠만 분포 – 韩语" lang="ko" hreflang="ko" data-title="볼츠만 분포" data-language-autonym="한국어" data-language-local-name="韩语" 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/Boltzmann-verdeling" title="Boltzmann-verdeling – 荷兰语" lang="nl" hreflang="nl" data-title="Boltzmann-verdeling" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Boltzmann-fordeling" title="Boltzmann-fordeling – 书面挪威语" lang="nb" hreflang="nb" data-title="Boltzmann-fordeling" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_Boltzmanna" title="Rozkład Boltzmanna – 波兰语" lang="pl" hreflang="pl" data-title="Rozkład Boltzmanna" data-language-autonym="Polski" data-language-local-name="波兰语" 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/Distribui%C3%A7%C3%A3o_de_Boltzmann" title="Distribuição de Boltzmann – 葡萄牙语" lang="pt" hreflang="pt" data-title="Distribuição de Boltzmann" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0" title="Распределение Больцмана – 俄语" lang="ru" hreflang="ru" data-title="Распределение Больцмана" data-language-autonym="Русский" data-language-local-name="俄语" 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/Bolcmanov_zakon_raspodele" title="Bolcmanov zakon raspodele – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Bolcmanov zakon raspodele" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Boltzmannovo_rozdelenie" title="Boltzmannovo rozdelenie – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Boltzmannovo rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Boltzmannf%C3%B6rdelning" title="Boltzmannfördelning – 瑞典语" lang="sv" hreflang="sv" data-title="Boltzmannfördelning" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boltzmann_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Boltzmann dağılımı – 土耳其语" lang="tr" hreflang="tr" data-title="Boltzmann dağılımı" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0" title="Розподіл Больцмана – 乌克兰语" lang="uk" hreflang="uk" data-title="Розподіл Больцмана" data-language-autonym="Українська" data-language-local-name="乌克兰语" 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/Boltzmann_taqsimoti" title="Boltzmann taqsimoti – 乌兹别克语" lang="uz" hreflang="uz" data-title="Boltzmann taqsimoti" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%8E%BB%E8%8C%B2%E6%9B%BC%E5%88%86%E4%BD%88" title="玻茲曼分佈 – 粤语" lang="yue" hreflang="yue" data-title="玻茲曼分佈" 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class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-5f97c608" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">（重定向自<a href="/w/index.php?title=%E6%B3%A2%E8%8C%B2%E6%9B%BC%E5%88%86%E5%B8%83&redirect=no" class="mw-redirect" title="波茲曼分布">波茲曼分布</a>）</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><div id="noteTA-5f97c608" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Physics"></div></div></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Exponential_probability_density.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Exponential_probability_density.svg/220px-Exponential_probability_density.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Exponential_probability_density.svg/330px-Exponential_probability_density.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Exponential_probability_density.svg/440px-Exponential_probability_density.svg.png 2x" data-file-width="360" data-file-height="288" /></a><figcaption>波茲曼分布是一種指數分布。</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Boltzmann_distribution_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Boltzmann_distribution_graph.svg/220px-Boltzmann_distribution_graph.svg.png" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Boltzmann_distribution_graph.svg/330px-Boltzmann_distribution_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Boltzmann_distribution_graph.svg/440px-Boltzmann_distribution_graph.svg.png 2x" data-file-width="480" data-file-height="340" /></a><figcaption>波茲曼因子 <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 p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> / <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 p_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499e0821b28c43e9bc2a6360b937de535057bc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.169ex; height:2.343ex;" alt="{\displaystyle p_{j}}"></span>(縱軸) 作為幾個不同能量差 <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}-\epsilon _{j}}"> <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> <mo>−</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{i}-\epsilon _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21f7ca131be8ed04aba44db1e8d7ed3a4bd08c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.438ex; height:2.676ex;" alt="{\displaystyle \epsilon _{i}-\epsilon _{j}}"></span> 的溫度 T 的函數</figcaption></figure> <p>在<a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6" title="统计力学">統計力學</a>和<a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">數學</a>中，<b>波茲曼分布</b>（英語：<span lang="en">Boltzmann distribution</span>），或稱<b>吉布斯分布</b>（英語：<span lang="en">Gibbs distribution</span>）<sup id="cite_ref-landau_1-0" class="reference"><a href="#cite_note-landau-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>，是一種<a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E5%B8%83" title="概率分布">機率分布</a>或<a href="/wiki/%E6%A6%82%E7%8E%87%E6%B5%8B%E5%BA%A6" title="概率测度">機率測度</a>，它給出一個系統處於某種狀態的機率，是該狀態的能量及溫度的函數。該分布以下列形式表示： </p> <dl><dd><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 p_{i}\propto e^{-{\varepsilon _{i}}/{(kT)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∝</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}\propto e^{-{\varepsilon _{i}}/{(kT)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaafd6c80731e68f1b903b782c2e4961f6e6ba1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.257ex; height:3.176ex;" alt="{\displaystyle p_{i}\propto e^{-{\varepsilon _{i}}/{(kT)}}}"></span></dd></dl> <p>其中<style data-mw-deduplicate="TemplateStyles:r58896141">.mw-parser-output .serif{font-family:Times,serif}</style><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">p<sub>i</sub></span></span>是系統處於狀態<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">i</span></span>的機率，<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ε<sub>i</sub></span></span>是該狀態的能量，<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">kT</span></span>為<a href="/wiki/%E6%B3%A2%E8%8C%B2%E6%9B%BC%E5%B8%B8%E6%95%B8" title="波茲曼常數">波茲曼常数</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">k</span></span>和<a href="/wiki/%E7%83%AD%E5%8A%9B%E5%AD%A6%E6%B8%A9%E5%BA%A6" title="热力学温度">熱力學溫度</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">T</span></span>的乘积。符號<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="{\textstyle \propto }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∝</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \propto }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddcdc1779a726be27007b9ef99ea3eb66680216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.676ex;" alt="{\textstyle \propto }"></span>表示<a href="/wiki/%E6%AF%94%E4%BE%8B" title="比例">比例</a>（比例常數見<a href="#分布形式">§ 分布形式</a>） </p><p><br /> 這裡的「系統」一詞具有非常廣泛的涵義；它適用的範圍可以從「足夠數量」的原子集合（但不是單個原子）到一個宏觀系統，例如天然气储罐。因此，波茲曼分布可以解決非常廣泛且多樣的問題。該分布表明，能量較低的狀態總是有較高的機率被佔用。 </p><p>兩種狀態的機率比稱為<b>波茲曼因子</b>，其特徵在於其僅取決於兩狀態之能量差： </p> <dl><dd><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 {\frac {p_{i}}{p_{j}}}=e^{{(\varepsilon _{j}-\varepsilon _{i})}/{(kT)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{i}}{p_{j}}}=e^{{(\varepsilon _{j}-\varepsilon _{i})}/{(kT)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae359810e14e71b62a886b2d6d38a067b06ad3b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.874ex; height:5.509ex;" alt="{\displaystyle {\frac {p_{i}}{p_{j}}}=e^{{(\varepsilon _{j}-\varepsilon _{i})}/{(kT)}}}"></span></dd></dl> <p>波茲曼分布以<a href="/wiki/%E8%B7%AF%E5%BE%B7%E7%BB%B4%E5%B8%8C%C2%B7%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC" title="路德维希·玻尔兹曼">路德維希·波茲曼</a>的名字命名，他於1868年研究<a href="/wiki/%E7%86%B1%E5%B9%B3%E8%A1%A1" title="熱平衡">熱平衡</a>中氣體的<a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6" title="统计力学">統計力學</a>時首次提出了這一分布。<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>波茲曼的統計力學成果證明於他的論文“論熱力學第二定律與熱平衡狀態的機率之間的關係”<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>該分布後來被<a href="/wiki/%E4%B9%94%E8%B5%9B%E4%BA%9A%C2%B7%E5%A8%81%E6%8B%89%E5%BE%B7%C2%B7%E5%90%89%E5%B8%83%E6%96%AF" title="乔赛亚·威拉德·吉布斯">喬賽亞·威拉德·吉布斯</a>（<span lang="en">Josiah Willard Gibbs</span>）以現代通用的形式進行了廣泛的研究。<sup id="cite_ref-gibbs_4-0" class="reference"><a href="#cite_note-gibbs-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>廣義波茲曼分布是<a href="/wiki/%E7%86%B5" title="熵">熵</a>的統計力學定義（<a href="/wiki/%E7%86%B5_(%E7%BB%9F%E8%AE%A1%E7%89%A9%E7%90%86%E5%AD%A6)" title="熵 (统计物理学)">吉布斯熵公式</a><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="{\textstyle S=-k_{\mathrm {B} }\sum _{i}p_{i}\log p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>log</mi> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S=-k_{\mathrm {B} }\sum _{i}p_{i}\log p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/154cde9bd6edcdc9c8066cebc798bf626d90d507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.725ex; height:3.009ex;" alt="{\textstyle S=-k_{\mathrm {B} }\sum _{i}p_{i}\log p_{i}}"></span>）和<a href="/wiki/%E7%86%B5" title="熵">熵</a>的熱力學定義（<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="{\textstyle \mathrm {d} S={\frac {\delta Q_{\text{rev}}}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rev</mtext> </mrow> </msub> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathrm {d} S={\frac {\delta Q_{\text{rev}}}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7356d80aa3a171878d42f8cdd3494d664c991a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.753ex; height:4.009ex;" alt="{\textstyle \mathrm {d} S={\frac {\delta Q_{\text{rev}}}{T}}}"></span>，以及<a href="/wiki/%E7%83%AD%E5%8A%9B%E5%AD%A6%E5%9F%BA%E6%9C%AC%E5%85%B3%E7%B3%BB" title="热力学基本关系">熱力學基本關係</a>）等價的充分必要條件。<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>不應將波茲曼分布与<a href="/wiki/%E9%BA%A6%E5%85%8B%E6%96%AF%E9%9F%A6-%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83" title="麦克斯韦-玻尔兹曼分布">馬克士威-波茲曼分布</a>或<a href="/wiki/%E9%BA%A6%E5%85%8B%E6%96%AF%E9%9F%A6-%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E7%BB%9F%E8%AE%A1" title="麦克斯韦-玻尔兹曼统计">馬克士威-波茲曼統計</a>混淆。波茲曼分布給出了系統處於某一狀態的機率，作為該狀態的能量的函數，<sup id="cite_ref-Atkins,_P._W._2010_6-0" class="reference"><a href="#cite_note-Atkins,_P._W._2010-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>而馬克士威-波茲曼分布給出了理想氣體中的粒子速度或能量的機率。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="分布形式"><span id=".E5.88.86.E5.B8.83.E5.BD.A2.E5.BC.8F"></span>分布形式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&action=edit&section=1" title="编辑章节：分布形式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>波茲曼分布是狀態能量與<a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>溫度的<a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E5%B8%83" title="概率分布">機率分布</a>函數，給出了粒子處於特定狀態下的機率<sup id="cite_ref-McQuarrie,_A._2000_7-0" class="reference"><a href="#cite_note-McQuarrie,_A._2000-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>。其具有以下形式： </p> <dl><dd><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 p_{i}={\frac {1}{Q}}}{e^{-{\varepsilon }_{i}/(kT)}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}"> <semantics> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Q</mi> </mfrac> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}={\frac {1}{Q}}}{e^{-{\varepsilon }_{i}/(kT)}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92a708962c080953dcf1d9e01435fc9294695c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-left: -0.089ex; width:33.907ex; height:7.343ex;" alt="{\displaystyle p_{i}={\frac {1}{Q}}}{e^{-{\varepsilon }_{i}/(kT)}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}"></span></dd></dl> <p>其中<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 p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span>為狀態i的機率，<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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be3140164b763359077d92b2cd33798eb6a488c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.744ex; height:2.009ex;" alt="{\displaystyle \epsilon _{i}}"></span>為狀態i之能量， <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>為波茲曼常數，<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> </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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>為系統的絕對溫度，而<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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>是系統中我們有興趣且可知的狀態數量。<sup id="cite_ref-McQuarrie,_A._2000_7-1" class="reference"><a href="#cite_note-McQuarrie,_A._2000-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Atkins,_P._W._2010_6-1" class="reference"><a href="#cite_note-Atkins,_P._W._2010-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>分母的歸一化常數<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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>（一些作者用<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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>表示）對系統所有狀態進行總和，是<a href="/wiki/%E9%85%8D%E5%88%86%E5%87%BD%E6%95%B0" title="配分函数">規範的配分函數</a>： </p> <dl><dd><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 Q={\sum _{i=1}^{M}{e^{-{\varepsilon }_{i}/(kT)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\sum _{i=1}^{M}{e^{-{\varepsilon }_{i}/(kT)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92543bdbf256f77c50f3794dc8bd25f27ea96141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.779ex; height:7.343ex;" alt="{\displaystyle Q={\sum _{i=1}^{M}{e^{-{\varepsilon }_{i}/(kT)}}}}"></span></dd></dl> <p>這個結果源自於所有可能狀態的機率之和必須為1的約束條件。 </p><p>波茲曼分布是使<a href="/wiki/%E7%86%B5" title="熵">熵</a>最大化的分布。 </p> <dl><dd><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 H(p_{1},p_{2},\cdots ,p_{M})=-\sum _{i=1}^{M}p_{i}\log _{2}p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(p_{1},p_{2},\cdots ,p_{M})=-\sum _{i=1}^{M}p_{i}\log _{2}p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e666e84d1aeb5a23138b69f1a5f40d7a7c14e97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.435ex; height:7.343ex;" alt="{\displaystyle H(p_{1},p_{2},\cdots ,p_{M})=-\sum _{i=1}^{M}p_{i}\log _{2}p_{i}}"></span></dd></dl> <p>受制於約束條件時，<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="{\textstyle \sum {p_{i}{\varepsilon }_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum {p_{i}{\varepsilon }_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb77fdfd0efb6a09bfd68a803e8c452789327a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.693ex; height:2.843ex;" alt="{\textstyle \sum {p_{i}{\varepsilon }_{i}}}"></span>等於特定的平均能量值（可以使用<a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B9%98%E6%95%B0" title="拉格朗日乘数">拉格朗日乘數</a>證明）。 </p><p>對於一個我們感興趣的系統，若是知道系統中各狀態的能量，可以直接計算此系統的配分函數。各種原子的配分函數可以在NIST Atomic Spectra Database找到。<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>該分布表明，低能量的狀態比起高能量的狀態具有較高的分布機率。同時，它也能夠定量地比較兩能階分布機率的關係。狀態i與狀態j的分布機率比為： </p> <dl><dd><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 {\frac {p_{i}}{p_{j}}}=e^{({\varepsilon }_{j}-{\varepsilon }_{i})/(kT)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{i}}{p_{j}}}=e^{({\varepsilon }_{j}-{\varepsilon }_{i})/(kT)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9e2acd760db0b83548a0edc336305daf8124a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.874ex; height:5.509ex;" alt="{\displaystyle {\frac {p_{i}}{p_{j}}}=e^{({\varepsilon }_{j}-{\varepsilon }_{i})/(kT)}}"></span></dd></dl> <p>其中，<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 p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span>為狀態i的機率，<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 p_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499e0821b28c43e9bc2a6360b937de535057bc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.169ex; height:2.343ex;" alt="{\displaystyle p_{j}}"></span>為狀態j的機率，而<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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be3140164b763359077d92b2cd33798eb6a488c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.744ex; height:2.009ex;" alt="{\displaystyle \epsilon _{i}}"></span>和<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 _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d007888875e2294ff7ef04b13349b27ec46e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.854ex; height:2.343ex;" alt="{\displaystyle \epsilon _{j}}"></span>分別為狀態i和狀態j的能量。兩能量對應的機率比，必須考慮它們的<a href="/wiki/%E7%AE%80%E5%B9%B6%E8%83%BD%E7%BA%A7" title="简并能级">簡併能階</a>。 </p><p>波茲曼分布通常用於描述粒子的分布，例如原子與分子在各種束縛態的分布情形。實際上，粒子處於狀態i的機率會等於處於狀態i的粒子數除以系統中所有粒子的總數，即： </p> <dl><dd><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 p_{i}={\frac {N_{i}}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}={\frac {N_{i}}{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/070b73bfce39fc38087a02ce97c1a66551a3741d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.659ex; height:5.176ex;" alt="{\displaystyle p_{i}={\frac {N_{i}}{N}}}"></span></dd></dl> <p>其中<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> </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/fef58cebf23adff9199f17325aefb5515fdca99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.666ex; height:2.509ex;" alt="{\displaystyle N_{i}}"></span>為處於狀態i的粒子數，<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>為系統中所有粒子的總數。我們可以使用波茲曼分布找出該機率。正如上式，機率等於位於狀態i的粒子數與總數之比例。因此，我們可以位於狀態i的粒子數比例表示成一以能量作為變數的函數：<sup id="cite_ref-Atkins,_P._W._2010_6-2" class="reference"><a href="#cite_note-Atkins,_P._W._2010-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\frac {N_{i}}{N}}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {N_{i}}{N}}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e98caf4fe7c8a7527085a0780ff34f078c58d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:21.477ex; height:7.343ex;" alt="{\displaystyle {\frac {N_{i}}{N}}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}"></span></dd></dl> <p>這個等式對於<a href="/wiki/%E5%85%89%E8%B0%B1%E5%AD%A6" title="光谱学">光譜學</a>來說非常重要。在光譜學中，我們觀察到一個原子或分子從一狀態躍遷至另一狀態的<a href="/wiki/%E8%AD%9C%E7%B7%9A" title="譜線">譜線</a>。<sup id="cite_ref-Atkins,_P._W._2010_6-3" class="reference"><a href="#cite_note-Atkins,_P._W._2010-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>一般來說，越大比例的分子在第一能態，意味著發生越多的從第一至第二能態的躍遷。此現象可從越強的譜線觀察到。然而，除了分子數比例外，也有其他因素會影響譜線的強弱，例如<a href="/wiki/%E7%A6%81%E5%88%B6%E6%A9%9F%E5%88%B6" title="禁制機制">禁制機制</a>。 </p><p>機器學習中常用的<a href="/wiki/Softmax%E5%87%BD%E6%95%B0" title="Softmax函数">softmax函數</a>與波茲曼函數有關。 </p> <div class="mw-heading mw-heading2"><h2 id="統計力學上的應用"><span id=".E7.B5.B1.E8.A8.88.E5.8A.9B.E5.AD.B8.E4.B8.8A.E7.9A.84.E6.87.89.E7.94.A8"></span>統計力學上的應用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&action=edit&section=2" title="编辑章节：統計力學上的應用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%AD%A3%E5%88%99%E7%B3%BB%E7%BB%BC" title="正则系综">正则系综</a>和<a href="/wiki/%E9%BA%A6%E5%85%8B%E6%96%AF%E9%9F%A6-%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E7%BB%9F%E8%AE%A1" title="麦克斯韦-玻尔兹曼统计">麦克斯韦-玻尔兹曼统计</a></div> <p>在<a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6" title="统计力学">統計力學</a>中，波茲曼分布會出現在<a href="/wiki/%E7%86%B1%E5%B9%B3%E8%A1%A1" title="熱平衡">熱平衡</a>（能量交換平衡）的孤立（或近似孤立）系統中。最一般的情況是正則系綜的機率分布。而在某些特殊情況下（衍生自正則系綜）也有相關的應用。 </p> <dl><dt><a href="/wiki/%E6%AD%A3%E5%88%99%E7%B3%BB%E7%BB%BC" title="正则系综">正則系綜</a>（一般情况）</dt> <dd><a href="/wiki/%E6%AD%A3%E5%88%99%E7%B3%BB%E7%BB%BC" title="正则系综">正則系綜</a>給出了各種可能狀態的<a href="/wiki/%E6%A9%9F%E7%8E%87" class="mw-redirect" title="機率">機率</a>分布在一固定體積、封閉且有<a href="/wiki/%E7%86%B1%E5%BA%AB" title="熱庫">熱庫</a>的熱平衡系統。此時，正則系綜具有波茲曼形式的機率分布。</dd></dl> <div class="mw-heading mw-heading2"><h2 id="數學上的應用"><span id=".E6.95.B8.E5.AD.B8.E4.B8.8A.E7.9A.84.E6.87.89.E7.94.A8"></span>數學上的應用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&action=edit&section=3" title="编辑章节：數學上的應用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/w/index.php?title=%E5%90%89%E5%B8%83%E6%96%AF%E6%B8%AC%E5%BA%A6&action=edit&redlink=1" class="new" title="吉布斯測度（页面不存在）">吉布斯測度</a>和<a href="/w/index.php?title=%E5%B0%8D%E6%95%B8-%E7%B7%9A%E6%80%A7%E6%A8%A1%E5%9E%8B&action=edit&redlink=1" class="new" title="對數-線性模型（页面不存在）">對數-線性模型</a></div> <p>在數學上，波茲曼函數更廣義的形式為<span class="ilh-all" data-orig-title="吉布斯測度" data-lang-code="en" data-lang-name="英语" data-foreign-title="Gibbs measure"><span class="ilh-page"><a href="/w/index.php?title=%E5%90%89%E5%B8%83%E6%96%AF%E6%B8%AC%E5%BA%A6&action=edit&redlink=1" class="new" title="吉布斯測度（页面不存在）">吉布斯測度</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Gibbs_measure" class="extiw" title="en:Gibbs measure"><span lang="en" dir="auto">Gibbs measure</span></a></span>）</span></span>。在<a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%AD%A6" title="统计学">統計學</a>與<a href="/wiki/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0" title="机器学习">機器學習</a>中又被稱為<span class="ilh-all" data-orig-title="對數-線性模型" data-lang-code="en" data-lang-name="英语" data-foreign-title="log-linear model"><span class="ilh-page"><a href="/w/index.php?title=%E5%B0%8D%E6%95%B8-%E7%B7%9A%E6%80%A7%E6%A8%A1%E5%9E%8B&action=edit&redlink=1" class="new" title="對數-線性模型（页面不存在）">對數-線性模型</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/log-linear_model" class="extiw" title="en:log-linear model"><span lang="en" dir="auto">log-linear model</span></a></span>）</span></span>。在深度学习中，玻尔兹曼分布被用于<a href="/wiki/%E9%9A%8F%E6%9C%BA%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C" title="随机神经网络">随机神经网络</a>的<a href="/w/index.php?title=%E9%87%87%E6%A0%B7%E5%88%86%E5%B8%83&action=edit&redlink=1" class="new" title="采样分布（页面不存在）">采样分布</a>，例如<a href="/wiki/%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E6%9C%BA" title="玻尔兹曼机">玻尔兹曼机</a>，<a href="/wiki/%E5%8F%97%E9%99%90%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E6%9C%BA" title="受限玻尔兹曼机">受限玻尔兹曼机</a>和<a href="/w/index.php?title=%E6%B7%B1%E5%BA%A6%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E6%9C%BA&action=edit&redlink=1" class="new" title="深度玻尔兹曼机（页面不存在）">深度玻尔兹曼机</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="參見"><span id=".E5.8F.83.E8.A6.8B"></span>參見</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&action=edit&section=4" title="编辑章节：參見"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E7%8E%BB%E8%89%B2%E2%80%93%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6%E7%BB%9F%E8%AE%A1" title="玻色–爱因斯坦统计">玻色–愛因斯坦統計</a></li> <li><a href="/wiki/%E8%B4%B9%E7%B1%B3-%E7%8B%84%E6%8B%89%E5%85%8B%E7%BB%9F%E8%AE%A1" title="费米-狄拉克统计">費米-狄拉克統計</a></li> <li><a href="/wiki/%E8%B4%9F%E6%B8%A9%E5%BA%A6" title="负温度">負溫度</a></li> <li><a href="/wiki/Softmax%E5%87%BD%E6%95%B0" title="Softmax函数">Softmax函数</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="參考文獻"><span id=".E5.8F.83.E8.80.83.E6.96.87.E7.8D.BB"></span>參考文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&action=edit&section=5" title="编辑章节：參考文獻"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-landau-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-landau_1-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/%E5%88%97%E5%A4%AB%C2%B7%E6%9C%97%E9%81%93" title="列夫·朗道">Landau, Lev Davidovich</a>; Lifshitz, Evgeny Mikhailovich. Statistical Physics. Course of Theoretical Physics <b>5</b> 3. Oxford: Pergamon Press. 1980 [1976]. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-7506-3372-7" title="Special:网络书源/0-7506-3372-7"><span title="国际标准书号">ISBN</span> 0-7506-3372-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&rft.au=Landau%2C+Lev+Davidovich&rft.au=Lifshitz%2C+Evgeny+Mikhailovich&rft.btitle=Statistical+Physics&rft.date=1980&rft.edition=3&rft.genre=book&rft.isbn=0-7506-3372-7&rft.place=Oxford&rft.pub=Pergamon+Press&rft.series=Course+of+Theoretical+Physics&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> Translated by J.B. Sykes and M.J. Kearsley. See section 28</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/%E8%B7%AF%E5%BE%B7%E7%BB%B4%E5%B8%8C%C2%B7%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC" title="路德维希·玻尔兹曼">Boltzmann, Ludwig</a>. Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten [Studies on the balance of living force between moving material points]. Wiener Berichte. 1868, <b>58</b>: 517–560.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&rft.atitle=Studien+%C3%BCber+das+Gleichgewicht+der+lebendigen+Kraft+zwischen+bewegten+materiellen+Punkten&rft.aufirst=Ludwig&rft.aulast=Boltzmann&rft.date=1868&rft.genre=article&rft.jtitle=Wiener+Berichte&rft.pages=517-560&rft.volume=58&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210305005604/http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf">Archived copy</a> <span style="font-size:85%;">(PDF)</span>. <span class="reference-accessdate"> [<span class="nowrap">2017-05-11</span>]</span>. （<a rel="nofollow" class="external text" href="http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf">原始内容</a> <span style="font-size:85%;">(PDF)</span>存档于2021-03-05）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&rft.btitle=Archived+copy&rft.genre=unknown&rft_id=http%3A%2F%2Fcrystal.med.upenn.edu%2Fsharp-lab-pdfs%2F2015SharpMatschinsky_Boltz1877_Entropy17.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-gibbs-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-gibbs_4-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Josiah_Willard_Gibbs" class="mw-redirect" title="Josiah Willard Gibbs">Gibbs, Josiah Willard</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/elementaryprinc02gibbgoog">Elementary Principles in Statistical Mechanics</a>. New York: <a href="/wiki/Charles_Scribner%27s_Sons" class="mw-redirect" title="Charles Scribner's Sons">Charles Scribner's Sons</a>. 1902.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&rft.aufirst=Josiah+Willard&rft.aulast=Gibbs&rft.btitle=Elementary+Principles+in+Statistical+Mechanics&rft.date=1902&rft.genre=book&rft.place=New+York&rft.pub=Charles+Scribner%27s+Sons&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementaryprinc02gibbgoog&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation journal">Gao, Xiang; Gallicchio, Emilio; Roitberg, Adrian. The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy. The Journal of Chemical Physics. 2019, <b>151</b> (3): 034113. <a rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/31325924"><span title="公共医学识别码">PMID 31325924</span></a>. <span class="plainlinks"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1903.02121"><span title="arXiv">arXiv:1903.02121</span></a> <span typeof="mw:File"><span title="可免费查阅"><img alt="可免费查阅" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>. <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.5111333"><span title="數位物件識別號">doi:10.1063/1.5111333</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&rft.atitle=The+generalized+Boltzmann+distribution+is+the+only+distribution+in+which+the+Gibbs-Shannon+entropy+equals+the+thermodynamic+entropy&rft.au=Gallicchio%2C+Emilio&rft.au=Roitberg%2C+Adrian&rft.aufirst=Xiang&rft.aulast=Gao&rft.date=2019&rft.genre=article&rft.issue=3&rft.jtitle=The+Journal+of+Chemical+Physics&rft.pages=034113&rft.volume=151&rft_id=info%3Aarxiv%2F1903.02121&rft_id=info%3Adoi%2F10.1063%2F1.5111333&rft_id=info%3Apmid%2F31325924&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Atkins,_P._W._2010-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Atkins,_P._W._2010_6-0"><sup><b>6.0</b></sup></a> <a href="#cite_ref-Atkins,_P._W._2010_6-1"><sup><b>6.1</b></sup></a> <a href="#cite_ref-Atkins,_P._W._2010_6-2"><sup><b>6.2</b></sup></a> <a href="#cite_ref-Atkins,_P._W._2010_6-3"><sup><b>6.3</b></sup></a></span> <span class="reference-text">Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York</span> </li> <li id="cite_note-McQuarrie,_A._2000-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-McQuarrie,_A._2000_7-0"><sup><b>7.0</b></sup></a> <a href="#cite_ref-McQuarrie,_A._2000_7-1"><sup><b>7.1</b></sup></a></span> <span class="reference-text"><cite class="citation book">McQuarrie, A. <a rel="nofollow" class="external text" href="https://archive.org/details/statisticalmecha00mcqu_0">Statistical Mechanics</a>. Sausalito, CA: University Science Books. 2000. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/1-891389-15-7" title="Special:网络书源/1-891389-15-7"><span title="国际标准书号">ISBN</span> 1-891389-15-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&rft.aufirst=A.&rft.aulast=McQuarrie&rft.btitle=Statistical+Mechanics&rft.date=2000&rft.genre=book&rft.isbn=1-891389-15-7&rft.place=Sausalito%2C+CA&rft.pub=University+Science+Books&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstatisticalmecha00mcqu_0&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://physics.nist.gov/PhysRefData/ASD/levels_form.html">NIST Atomic Spectra Database Levels Form</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20110320190125/http://physics.nist.gov/PhysRefData/ASD/levels_form.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） at nist.gov</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation book">Atkins, P. W.; de Paula, J. Physical Chemistry 9th. Oxford: Oxford University Press. 2009. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-19-954337-3" title="Special:网络书源/978-0-19-954337-3"><span title="国际标准书号">ISBN</span> 978-0-19-954337-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%88%86%E5%B8%83&rft.au=de+Paula%2C+J.&rft.aufirst=P.+W.&rft.aulast=Atkins&rft.btitle=Physical+Chemistry&rft.date=2009&rft.edition=9th&rft.genre=book&rft.isbn=978-0-19-954337-3&rft.place=Oxford&rft.pub=Oxford+University+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" 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