Bose-Einstein-Statistik

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id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Bose-einstein-fermi-dirac.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Bose-einstein-fermi-dirac.png/300px-Bose-einstein-fermi-dirac.png" decoding="async" width="300" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Bose-einstein-fermi-dirac.png/450px-Bose-einstein-fermi-dirac.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Bose-einstein-fermi-dirac.png/600px-Bose-einstein-fermi-dirac.png 2x" data-file-width="864" data-file-height="548" /></a><figcaption> <a href="/wiki/Besetzungszahl" class="mw-redirect" title="Besetzungszahl">Besetzungszahl</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="{\displaystyle \langle n\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591b81beaaf66f9e21fe885d4a9521c0390bee67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.204ex; height:2.843ex;" alt="{\displaystyle \langle n\rangle }"></span> als Funktion der Energie <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 E-\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>−</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E-\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ec9713bd5dd06fd1a1e4751e1a901577f6786a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.018ex; height:2.676ex;" alt="{\displaystyle E-\mu }"></span><br /> für Bosonen (<b>Bose-Einstein-Statistik, obere Kurve</b>)<br /> bzw. Fermionen (Fermi-Dirac-Statistik, untere Kurve),<br /> jeweils im Spezialfall der Wechselwirkungsfreiheit und bei konstanter Temperatur <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39e90b9e1e7d5be3b5eae57729dc63494bbe3fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T>0}"></span>.<br /> Das chemische Potential <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> </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/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> ist ein Parameter, der von Temperatur und Dichte abhängt;<br /> im Bose-Fall ist es immer kleiner als die Energie und würde im Grenzfall der <a href="/wiki/Bose-Einstein-Kondensation" class="mw-redirect" title="Bose-Einstein-Kondensation">Bose-Einstein-Kondensation</a> verschwinden;<br /> im Fermi-Fall dagegen ist es positiv, bei <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=0\,\mathrm {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=0\,\mathrm {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/404a1d4c65272bfa4bf9f2aa881bb3fea02f7aa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.092ex; height:2.176ex;" alt="{\displaystyle T=0\,\mathrm {K} }"></span> entspricht es der <a href="/wiki/Fermi-Energie" title="Fermi-Energie">Fermi-Energie</a>.</figcaption></figure> <p>Die <b>Bose-Einstein-Statistik</b> oder auch <b>Bose-Einstein-Verteilung</b>, benannt nach <a href="/wiki/Satyendranath_Bose" title="Satyendranath Bose">Satyendranath Bose</a> (1894–1974) und <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> (1879–1955), ist eine <a href="/wiki/Wahrscheinlichkeitsverteilung" class="mw-redirect" title="Wahrscheinlichkeitsverteilung">Wahrscheinlichkeitsverteilung</a> in der <a href="/wiki/Quantenstatistik" title="Quantenstatistik">Quantenstatistik</a> (<i>dort auch die Herleitung</i>). Sie beschreibt die mittlere <a href="/wiki/Besetzungszahl" class="mw-redirect" title="Besetzungszahl">Besetzungszahl</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="{\displaystyle \langle n(E)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n(E)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b7cdd4a3e81199348f2ab653659c2ddad50921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.789ex; height:2.843ex;" alt="{\displaystyle \langle n(E)\rangle }"></span> eines <a href="/wiki/Quantenzustand" class="mw-redirect" title="Quantenzustand">Quantenzustands</a> der Energie <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 E\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.163ex; height:2.176ex;" alt="{\displaystyle E\,}"></span> im <a href="/wiki/Thermodynamisches_Gleichgewicht" title="Thermodynamisches Gleichgewicht">thermodynamischen Gleichgewicht</a> bei der <a href="/wiki/Absolute_Temperatur" class="mw-redirect" title="Absolute Temperatur">absoluten Temperatur</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="{\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> für <a href="/wiki/Identische_Teilchen" class="mw-redirect" title="Identische Teilchen">identische</a> <a href="/wiki/Boson" title="Boson">Bosonen</a> als besetzende Teilchen. </p><p>Analog existiert für <a href="/wiki/Fermion" title="Fermion">Fermionen</a> die <a href="/wiki/Fermi-Dirac-Statistik" title="Fermi-Dirac-Statistik">Fermi-Dirac-Statistik</a>, die ebenso wie die Bose-Einstein-Statistik im Grenzfall großer Energie <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> in die <a href="/wiki/Boltzmann-Statistik" title="Boltzmann-Statistik">Boltzmann-Statistik</a> übergeht. </p><p>Kernpunkt der Bose-Einstein-Statistik ist, dass bei gleichzeitiger Vertauschung aller vier Variablen <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 x,y,z,m\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>m</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z,m\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ac1bd9e8a2387fe40afc52edf8b19d93f7c02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.103ex; height:2.009ex;" alt="{\displaystyle x,y,z,m\,}"></span> zweier Bosonen (<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 x,y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a00142f115caa7d735392675095459a5477afe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.906ex; height:2.009ex;" alt="{\displaystyle x,y\,}"></span> und <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> <mspace width="thinmathspace" /> </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/624faa61961bd63f364dee3e97dec7dd48694600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.475ex; height:1.676ex;" alt="{\displaystyle z\,}"></span>: Ortsvariable; <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> <mspace width="thinmathspace" /> </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/37129e832c2c81b9f146dd22228d409bd099b295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.427ex; height:1.676ex;" alt="{\displaystyle m\,}"></span>: <a href="/wiki/Spin" title="Spin">Spin</a>variable) die <a href="/wiki/Wellenfunktion" title="Wellenfunktion">Wellenfunktion</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="{\displaystyle \psi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0246f660f8317d29b9b2f21c339b3fe1171740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.9ex; height:2.509ex;" alt="{\displaystyle \psi \,}"></span> bzw. der <a href="/wiki/Zustandsvektor" class="mw-redirect" title="Zustandsvektor">Zustandsvektor</a> eines <a href="/wiki/Vielteilchensystem" class="mw-redirect" title="Vielteilchensystem">Vielteilchensystems</a> <i>nicht</i> das <a href="/wiki/Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> wechselt <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 (\psi \rightarrow \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\psi \rightarrow \psi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ac52d3a42a3ae64cf2993f75260bc6c998ac95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.45ex; height:2.843ex;" alt="{\displaystyle (\psi \rightarrow \psi )}"></span>, während es in der Fermi-Dirac-Statistik sehr wohl wechselt <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 (\psi \rightarrow -\psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\psi \rightarrow -\psi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acca2d49e516ee2ba2384f758bdc5b41b73d2d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.258ex; height:2.843ex;" alt="{\displaystyle (\psi \rightarrow -\psi )}"></span>. Im Gegensatz zu Fermionen können deshalb mehrere Bosonen im gleichen Ein-Teilchen-Zustand sein, also die gleichen <a href="/wiki/Quantenzahl" title="Quantenzahl">Quantenzahlen</a> haben. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Bei_Wechselwirkungsfreiheit"><span class="tocnumber">1</span> <span class="toctext">Bei Wechselwirkungsfreiheit</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Herleitung_aus_einem_Minimum_der_freien_Energie"><span class="tocnumber">2</span> <span class="toctext">Herleitung aus einem Minimum der freien Energie</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Literatur"><span class="tocnumber">3</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Einzelnachweise"><span class="tocnumber">4</span> <span class="toctext">Einzelnachweise</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Siehe_auch"><span class="tocnumber">5</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Weblinks"><span class="tocnumber">6</span> <span class="toctext">Weblinks</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Bei_Wechselwirkungsfreiheit">Bei Wechselwirkungsfreiheit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bose-Einstein-Statistik&veaction=edit&section=1" title="Abschnitt bearbeiten: Bei Wechselwirkungsfreiheit" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bose-Einstein-Statistik&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Bei Wechselwirkungsfreiheit"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei Wechselwirkungsfreiheit (<a href="/wiki/Bosegas" class="mw-redirect" title="Bosegas">Bosegas</a>) ergibt sich für Bosonen die folgende Formel: </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 \langle n(E)\rangle ={\frac {1}{e^{\beta (E-\mu )}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <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> <mi>E</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 \langle n(E)\rangle ={\frac {1}{e^{\beta (E-\mu )}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0e773767fd54842b0945b1fa8f095f930a6939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.789ex; height:5.676ex;" alt="{\displaystyle \langle n(E)\rangle ={\frac {1}{e^{\beta (E-\mu )}-1}}}"></span></dd></dl> <p>mit </p> <ul><li>dem <a href="/wiki/Chemisches_Potential" title="Chemisches Potential">chemischen Potential</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="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </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/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>, welches für Bosonen stets kleiner als der niedrigste mögliche Energiewert ist: <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 <E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo><</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu <E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c886d366f6c8fbdcd1c612abb142832a050a9598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.276ex; height:2.676ex;" alt="{\displaystyle \mu <E}"></span>;<br />daher ist die Bose-Einstein-Statistik nur für Energiewerte <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 E-\mu >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>−</mo> <mi>μ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E-\mu >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf9b98641ed66a377c3f5e0516a246572203be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.279ex; height:2.676ex;" alt="{\displaystyle E-\mu >0}"></span> definiert.</li> <li>der <i>Energienormierung</i> <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>. Die Wahl von <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> hängt von der verwendeten Temperaturskala ab: <ul><li>üblicherweise wird sie gewählt zu <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 \beta =1/(k_{\mathrm {B} }T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1/(k_{\mathrm {B} }T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c498f17a4ba8405ab3fa1396d47d46c9513c23aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.808ex; height:2.843ex;" alt="{\displaystyle \beta =1/(k_{\mathrm {B} }T)}"></span> mit der <a href="/wiki/Boltzmann-Konstante" title="Boltzmann-Konstante">Boltzmann-Konstanten</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="{\displaystyle k_{\mathrm {B} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\mathrm {B} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\mathrm {B} }}"></span>;</li> <li>sie beträgt <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 \beta =1/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1e21c8b3cc0966416fbb167dc0f257e83fd65e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.392ex; height:2.843ex;" alt="{\displaystyle \beta =1/T}"></span>, wenn die Temperatur in Energieeinheiten, etwa <a href="/wiki/Joule" title="Joule">Joule</a>, gemessen wird; dies geschieht, wenn <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_{\mathrm {B} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\mathrm {B} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\mathrm {B} }}"></span> auch in der Definition der <a href="/wiki/Entropie_(Thermodynamik)" class="mw-redirect" title="Entropie (Thermodynamik)">Entropie</a> – welche dann einheitenlos ist – nicht auftaucht.</li></ul></li></ul> <p>Unterhalb einer sehr tiefen kritischen Temperatur <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_{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/633755bffbd9959a3b5c9388a1c3567be9347649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.509ex;" alt="{\displaystyle T_{\lambda }}"></span> erhält man bei Wechselwirkungsfreiheit – unter der Annahme, dass <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> gegen das Energie-Minimum strebt – die <a href="/wiki/Bose-Einstein-Kondensation" class="mw-redirect" title="Bose-Einstein-Kondensation">Bose-Einstein-Kondensation</a>. </p><p>Man beachte, dass es sich bei <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 \langle n(E)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n(E)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b7cdd4a3e81199348f2ab653659c2ddad50921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.789ex; height:2.843ex;" alt="{\displaystyle \langle n(E)\rangle }"></span> um die Besetzungszahl eines Quantenzustandes handelt. Benötigt man die Besetzungszahl eines <a href="/wiki/Entartung_(Quantenmechanik)" title="Entartung (Quantenmechanik)">entarteten</a> <a href="/wiki/Energieniveau" title="Energieniveau">Energieniveaus</a>, so ist obiger Ausdruck zusätzlich mit dem entsprechenden <i>Entartungsgrad</i> <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}=2s+1}"> <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> <mo>=</mo> <mn>2</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}=2s+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/327253ff83fc7cbc79ba2ea29eb90edf5838bde5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.263ex; height:2.509ex;" alt="{\displaystyle g_{i}=2s+1}"></span> zu multiplizieren (<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 s\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c16c08f3f12ff4a2ebcda1059da2e917c75996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.478ex; height:1.676ex;" alt="{\displaystyle s\,}"></span>: Spin, bei Bosonen immer ganzzahlig), vgl. auch <a href="/wiki/Multiplizit%C3%A4t" title="Multiplizität">Multiplizität</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Herleitung_aus_einem_Minimum_der_freien_Energie">Herleitung aus einem Minimum der freien Energie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bose-Einstein-Statistik&veaction=edit&section=2" title="Abschnitt bearbeiten: Herleitung aus einem Minimum der freien Energie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bose-Einstein-Statistik&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Herleitung aus einem Minimum der freien Energie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aus der Bedingung, dass im thermischen Gleichgewicht (bei konstanter Temperatur <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>, Teilchenzahl <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> und Volumen <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>) die <a href="/wiki/Freie_Energie" title="Freie Energie">freie Energie</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 F=E-TS\qquad (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>E</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=E-TS\qquad (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d569849535f4f229ef67e3d19ceef695e11be201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.208ex; height:2.843ex;" alt="{\displaystyle F=E-TS\qquad (1)}"></span></dd></dl> <p>ein Minimum annimmt, kann die Bose-Einstein-Statistik mit wenigen Annahmen hergeleitet werden. Es sei <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> die Gesamtzahl aller Bosonen und <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> die Anzahl Bosonen im Energieniveau <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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span> mit <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 i=1,2,\dots ,I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2,\dots ,I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0c47cabb81ffa687d04e8f91e37d2bc9f00094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.61ex; height:2.509ex;" alt="{\displaystyle i=1,2,\dots ,I}"></span>, d. h. die <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> Bosonen seien über die Energieniveaus <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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span> verteilt. <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 D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span> sei die Anzahl der möglichen Zustände im Energieniveau <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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span>, d. h. die Energieniveaus <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 E_{1},E_{2},\dots ,E_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1},E_{2},\dots ,E_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/536f05de58356a65be8949b5906c3c91c9d4b8fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.527ex; height:2.509ex;" alt="{\displaystyle E_{1},E_{2},\dots ,E_{I}}"></span> seien jeweils <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 D_{1},D_{2},\dots ,D_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{1},D_{2},\dots ,D_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fb955e2e3aba1208440893489d2923397cf65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.154ex; height:2.509ex;" alt="{\displaystyle D_{1},D_{2},\dots ,D_{I}}"></span> -fach entartet. Für den Makrozustand des Systems ist es unerheblich, welche der <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> Bosonen sich im <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-ten Energieniveau befinden und welche der <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 D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span> Zustände sie darin besetzen. Der Makrozustand wird daher vollständig durch <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_{1},N_{2},\dots ,N_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{1},N_{2},\dots ,N_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db174d1dbc20012573e1fd2e1478567fd32089d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.98ex; height:2.509ex;" alt="{\displaystyle N_{1},N_{2},\dots ,N_{I}}"></span> bestimmt. </p><p>Für eine beliebige Verteilung der Bosonen auf die Energieniveaus gilt: </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 {\begin{aligned}N&=\sum _{i=1}^{I}N_{i}&\qquad (2)\\E&=\sum _{i=1}^{I}N_{i}E_{i}&\qquad (3)\\S&=k_{\rm {B}}\ln W.&\qquad (4)&\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>N</mi> </mtd> <mtd> <mi></mi> <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>I</mi> </mrow> </munderover> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>E</mi> </mtd> <mtd> <mi></mi> <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>I</mi> </mrow> </munderover> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mi>W</mi> <mo>.</mo> </mtd> <mtd> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}N&=\sum _{i=1}^{I}N_{i}&\qquad (2)\\E&=\sum _{i=1}^{I}N_{i}E_{i}&\qquad (3)\\S&=k_{\rm {B}}\ln W.&\qquad (4)&\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3230a259db9f4091f2d4ce1ae0981ff10d1e2204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:27.099ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}N&=\sum _{i=1}^{I}N_{i}&\qquad (2)\\E&=\sum _{i=1}^{I}N_{i}E_{i}&\qquad (3)\\S&=k_{\rm {B}}\ln W.&\qquad (4)&\end{aligned}}}"></span></dd></dl> <p>Gleichung (2) gibt die Gesamtzahl der Bosonen wieder, die konstant gehalten werden soll, während die einzelnen <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> variiert werden, um das Minimum von <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> zu erreichen. Gleichung (3) gibt die zur vorliegenden Verteilung gehörende Energie <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> des Systems an. Gleichung (4) ist (nach <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>) die Entropie des Zustands des Systems (Makrozustand), wobei <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 \textstyle W=\prod _{i=1}^{I}W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>W</mi> <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>I</mi> </mrow> </munderover> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle W=\prod _{i=1}^{I}W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26204352dae6f8c1677546549481f3b84d03a8a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.008ex; height:3.509ex;" alt="{\displaystyle \textstyle W=\prod _{i=1}^{I}W_{i}}"></span> die Wahrscheinlichkeit für die Besetzungszahlen <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_{1},N_{2},\dots ,N_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{1},N_{2},\dots ,N_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db174d1dbc20012573e1fd2e1478567fd32089d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.98ex; height:2.509ex;" alt="{\displaystyle N_{1},N_{2},\dots ,N_{I}}"></span> angibt, d. h. die Anzahl der möglichen Verteilungen (Mikrozustände) von jeweils <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> Bosonen auf die Plätze <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 D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span> für alle Energieniveaus <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 i=1,2,\dots ,I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2,\dots ,I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0c47cabb81ffa687d04e8f91e37d2bc9f00094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.61ex; height:2.509ex;" alt="{\displaystyle i=1,2,\dots ,I}"></span>. Aus Gleichung (4) folgt damit: </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 S=k_{\rm {B}}\ln \prod _{i=1}^{I}W_{i}=k_{\rm {B}}\sum _{i=1}^{I}\ln W_{i}.\qquad (5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>ln</mi> <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>I</mi> </mrow> </munderover> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=k_{\rm {B}}\ln \prod _{i=1}^{I}W_{i}=k_{\rm {B}}\sum _{i=1}^{I}\ln W_{i}.\qquad (5)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ddd62afa38afdc7c090268c3747112af8710db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.074ex; height:7.343ex;" alt="{\displaystyle S=k_{\rm {B}}\ln \prod _{i=1}^{I}W_{i}=k_{\rm {B}}\sum _{i=1}^{I}\ln W_{i}.\qquad (5)}"></span></dd></dl> <p>Dabei gibt der Binomialkoeffizient </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 W_{i}={\binom {N_{i}+D_{i}-1}{N_{i}}}={\frac {(N_{i}+D_{i}-1)!}{N_{i}!(D_{i}-1)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}={\binom {N_{i}+D_{i}-1}{N_{i}}}={\frac {(N_{i}+D_{i}-1)!}{N_{i}!(D_{i}-1)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5419ad80b632f9d98bd0755816078f1a69c8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.37ex; height:6.509ex;" alt="{\displaystyle W_{i}={\binom {N_{i}+D_{i}-1}{N_{i}}}={\frac {(N_{i}+D_{i}-1)!}{N_{i}!(D_{i}-1)!}}}"></span></dd></dl> <p>die Anzahl der Möglichkeiten an, dass <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> Teilchen ohne Beachtung der Reihenfolge das <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 D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span>-fach entartete Energieniveau <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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span> zu besetzen (<a href="/wiki/Kombination_(Kombinatorik)#Kombination_mit_Wiederholung" title="Kombination (Kombinatorik)">Kombination mit Wiederholung</a> von <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> Teilchen). </p><p>Mit Hilfe der beiden ersten dominierenden Glieder der <a href="/wiki/Stirlingformel" title="Stirlingformel">Stirling-Reihe</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="{\displaystyle \ln k!\approx k\ln k-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> <mo>!</mo> <mo>≈</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln k!\approx k\ln k-k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c653e37e8e19c64bf742b7e1a3816bb4ec56b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.471ex; height:2.343ex;" alt="{\displaystyle \ln k!\approx k\ln k-k}"></span> für <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\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\to \infty }"></span>) und <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}>>1,D_{i}>>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> <mo>>></mo> <mn>1</mn> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{i}>>1,D_{i}>>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f93fd759a59c74a5bc6f9186efb981fa51f9060d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.562ex; height:2.509ex;" alt="{\displaystyle N_{i}>>1,D_{i}>>1}"></span> ergibt sich weiter </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 {\begin{aligned}\ln W_{i}&=\ln(N_{i}+D_{i}-1)!-\ln N_{i}!-\ln(D_{i}-1)!\\&\approx (N_{i}+D_{i}-1)\ln(N_{i}+D_{i}-1)-(N_{i}+D_{i}-1)-N_{i}\ln N_{i}+N_{i}-(D_{i}-1)\ln(D_{i}-1)+(D_{i}-1)\\&=(N_{i}+D_{i}-1)\ln(N_{i}+D_{i}-1)-N_{i}\ln N_{i}-(D_{i}-1)\ln(D_{i}-1).\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (6)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln W_{i}&=\ln(N_{i}+D_{i}-1)!-\ln N_{i}!-\ln(D_{i}-1)!\\&\approx (N_{i}+D_{i}-1)\ln(N_{i}+D_{i}-1)-(N_{i}+D_{i}-1)-N_{i}\ln N_{i}+N_{i}-(D_{i}-1)\ln(D_{i}-1)+(D_{i}-1)\\&=(N_{i}+D_{i}-1)\ln(N_{i}+D_{i}-1)-N_{i}\ln N_{i}-(D_{i}-1)\ln(D_{i}-1).\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (6)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184c69a5addf3cb8f157864f479bc13165824f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:113.873ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\ln W_{i}&=\ln(N_{i}+D_{i}-1)!-\ln N_{i}!-\ln(D_{i}-1)!\\&\approx (N_{i}+D_{i}-1)\ln(N_{i}+D_{i}-1)-(N_{i}+D_{i}-1)-N_{i}\ln N_{i}+N_{i}-(D_{i}-1)\ln(D_{i}-1)+(D_{i}-1)\\&=(N_{i}+D_{i}-1)\ln(N_{i}+D_{i}-1)-N_{i}\ln N_{i}-(D_{i}-1)\ln(D_{i}-1).\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (6)\\\end{aligned}}}"></span></dd></dl> <p>Um die Verteilung zu finden, bei der die freie Energie <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> unter der Nebenbedingung <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=\mathrm {const} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\mathrm {const} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/484d83b91f28b893aca48904620ae22ae60aae62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.47ex; height:2.176ex;" alt="{\displaystyle N=\mathrm {const} }"></span>, aber <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> variabel, minimal wird, kann die Methode der <a href="/wiki/Lagrange-Multiplikator" title="Lagrange-Multiplikator">Lagrange-Multiplikatoren</a> benutzt werden: </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 {\partial F}{\partial N_{i}}}-\lambda {\frac {\partial N}{\partial N_{i}}}=0\qquad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>N</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial F}{\partial N_{i}}}-\lambda {\frac {\partial N}{\partial N_{i}}}=0\qquad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ba135f404823236293ff0ccb29498c866b991b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.742ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial F}{\partial N_{i}}}-\lambda {\frac {\partial N}{\partial N_{i}}}=0\qquad }"></span> für <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 i=1,2,3\dots I.\qquad (7)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>…</mo> <mi>I</mi> <mo>.</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2,3\dots I.\qquad (7)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba91901d4f1eb529cc843856cac098ad679d2119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.776ex; height:2.843ex;" alt="{\displaystyle i=1,2,3\dots I.\qquad (7)}"></span></dd></dl> <p>Darin ist <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> der (von <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> unabhängige) Lagrange-Multiplikator. Hierbei gilt für die partiellen Ableitungen <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 {\partial N}{\partial N_{i}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>N</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial N}{\partial N_{i}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130caf41f6a836078f49ec1eebb9061e078fd764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.081ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial N}{\partial N_{i}}}=1}"></span> und <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 {\partial E}{\partial N_{i}}}=E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial E}{\partial N_{i}}}=E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72d89fa2e6488a3337b69e68ef9e0511b66b62eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.433ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial E}{\partial N_{i}}}=E_{i}}"></span>, da jedes <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> genau einmal linear in der Summe von Gleichung (2) bzw. (3) vorkommt. Da <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> nur Variable von <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 W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span>, aber nicht von <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 W_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa98874e6beb16373e8d0e056ba550cf653676a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.103ex; height:2.843ex;" alt="{\displaystyle W_{j}}"></span> mit <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 j\neq i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\neq i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b7538abe3f69348eb371e48d7b33a8994610e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:4.886ex; height:2.676ex;" alt="{\displaystyle j\neq i}"></span> ist, vereinfacht sich die Summe von Gleichung (5) durch die partielle Ableitung nach <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> wie folgt: <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 {\partial S}{\partial N_{i}}}=k_{\rm {B}}{\frac {\partial \ln W_{i}}{\partial N_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial S}{\partial N_{i}}}=k_{\rm {B}}{\frac {\partial \ln W_{i}}{\partial N_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f96b6f5bcc59afbdb86f456f044f5e0f7592fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.387ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial S}{\partial N_{i}}}=k_{\rm {B}}{\frac {\partial \ln W_{i}}{\partial N_{i}}}}"></span>. </p><p>Damit ergibt sich aus Gleichung (1) und (7): </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 \lambda ={\frac {\partial F}{\partial N_{i}}}={\frac {\partial E}{\partial N_{i}}}-T{\frac {\partial S}{\partial N_{i}}}=E_{i}-k_{\rm {B}}T{\frac {\partial \ln W_{i}}{\partial N_{i}}}.\qquad (8)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {\partial F}{\partial N_{i}}}={\frac {\partial E}{\partial N_{i}}}-T{\frac {\partial S}{\partial N_{i}}}=E_{i}-k_{\rm {B}}T{\frac {\partial \ln W_{i}}{\partial N_{i}}}.\qquad (8)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2835b63e7497201f9982acd6c34df5f1a2df3e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.698ex; height:5.843ex;" alt="{\displaystyle \lambda ={\frac {\partial F}{\partial N_{i}}}={\frac {\partial E}{\partial N_{i}}}-T{\frac {\partial S}{\partial N_{i}}}=E_{i}-k_{\rm {B}}T{\frac {\partial \ln W_{i}}{\partial N_{i}}}.\qquad (8)}"></span></dd></dl> <p><br /> Die partielle Ableitung <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 {\partial \ln W_{i}}{\partial N_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \ln W_{i}}{\partial N_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3b295b85fad318bfe1fd02d6e2e803854dfbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.861ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \ln W_{i}}{\partial N_{i}}}}"></span> kann aus Gl. (6) mit <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}>>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> <mo>>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{i}>>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1db8e73fc0a80641ae0dc7253ef942ce54d0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.735ex; height:2.509ex;" alt="{\displaystyle N_{i}>>1}"></span> berechnet werden: </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 {\begin{aligned}{\frac {\partial \ln W_{i}}{\partial N_{i}}}&\approx \ln(N_{i}+D_{i}-1)+1-\ln N_{i}-1\\&=\ln(N_{i}+D_{i}-1)-\ln N_{i}\\&=\ln(D_{i}/N_{i}+1-1/N_{i})\\&\approx \ln(D_{i}/N_{i}+1).\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial \ln W_{i}}{\partial N_{i}}}&\approx \ln(N_{i}+D_{i}-1)+1-\ln N_{i}-1\\&=\ln(N_{i}+D_{i}-1)-\ln N_{i}\\&=\ln(D_{i}/N_{i}+1-1/N_{i})\\&\approx \ln(D_{i}/N_{i}+1).\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f6204d646718b433c006cfaced80225aeb7f288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:43.531ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial \ln W_{i}}{\partial N_{i}}}&\approx \ln(N_{i}+D_{i}-1)+1-\ln N_{i}-1\\&=\ln(N_{i}+D_{i}-1)-\ln N_{i}\\&=\ln(D_{i}/N_{i}+1-1/N_{i})\\&\approx \ln(D_{i}/N_{i}+1).\\\end{aligned}}}"></span></dd></dl> <p>Damit ergibt sich aus Gleichung (8) </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 \lambda =E_{i}-k_{\rm {B}}T\ln(D_{i}/N_{i}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =E_{i}-k_{\rm {B}}T\ln(D_{i}/N_{i}+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fcac76835b06a27fbca6b2fb34740eb03d3af91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.743ex; height:2.843ex;" alt="{\displaystyle \lambda =E_{i}-k_{\rm {B}}T\ln(D_{i}/N_{i}+1)}"></span>.</dd></dl> <p>Einsetzen der durch <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 f_{i}:={\frac {N_{i}}{D_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</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> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}:={\frac {N_{i}}{D_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02507da98574cdf8152a4722f6b9549b4969d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.244ex; height:5.676ex;" alt="{\displaystyle f_{i}:={\frac {N_{i}}{D_{i}}}}"></span> gegebenen Besetzungswahrscheinlichkeit <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 f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"></span> und Umstellung ergibt </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 f_{i}={\frac {1}{\exp {\frac {E_{i}-\lambda }{k_{\rm {B}}T}}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>λ</mi> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}={\frac {1}{\exp {\frac {E_{i}-\lambda }{k_{\rm {B}}T}}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b26f1d7552a43c90e729fe33e0f6c8828e199a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:18.727ex; height:7.509ex;" alt="{\displaystyle f_{i}={\frac {1}{\exp {\frac {E_{i}-\lambda }{k_{\rm {B}}T}}-1}}}"></span>.</dd></dl> <p>Dies ist die Bose-Einstein-Statistik. Der Lagrange-Multiplikator erweist sich als ihr chemisches Potential <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 =\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c3f5ceae83a5ff1133cf5d3a55630554cd05ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.855ex; height:2.676ex;" alt="{\displaystyle \mu =\lambda }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bose-Einstein-Statistik&veaction=edit&section=3" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bose-Einstein-Statistik&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>U. Krey, A. Owen: <i>Basic Theoretical Physics – a Concise Overview</i>. Springer, Berlin//Heidelberg/New York 2007, <a href="/wiki/Spezial:ISBN-Suche/9783540368045" class="internal mw-magiclink-isbn">ISBN 978-3-540-36804-5</a> (auf Englisch).</li> <li>L. D. Landau, E. M. Lifschitz: <i>Statistische Physik</i>. Verlag Harri Deutsch (ehem. Akademie Verlag), Berlin 1987 (verwendet unübliche Temperatureinheit).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bose-Einstein-Statistik&veaction=edit&section=4" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bose-Einstein-Statistik&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Walter Greiner, Ludwig Neise, <a href="/wiki/Horst_St%C3%B6cker" title="Horst Stöcker">Horst Stöcker</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Thermodynamics and Statistical Mechanics</cite>. Springer-Verlag, Berlin, Heidelberg, New York 1997, <a href="/wiki/Spezial:ISBN-Suche/0387942998" class="internal mw-magiclink-isbn">ISBN 0-387-94299-8</a> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bose-Einstein-Statistik&rft.au=Walter+Greiner%2C+Ludwig+Neise%2C+Horst+St%C3%B6cker&rft.btitle=Thermodynamics+and+Statistical+Mechanics&rft.date=1997&rft.genre=book&rft.isbn=0387942998&rft.place=Berlin%2C+Heidelberg%2C+New+York&rft.pub=Springer-Verlag" style="display:none"> </span> insbes. S. 310–313</li></ul> <ul><li>Marcelo Alonso, Edward J. Finn: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Fundamental University Physics, Vol. III, Quantum and Statistical Physics</cite>. Addison-Wesley Publishing Company, Massachusetts 1968 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bose-Einstein-Statistik&rft.au=Marcelo+Alonso%2C+Edward+J.+Finn&rft.btitle=Fundamental+University+Physics%2C+Vol.+III%2C+Quantum+and+Statistical+Physics&rft.date=1968&rft.genre=book&rft.place=Massachusetts&rft.pub=Addison-Wesley+Publishing+Company" style="display:none"> </span>QRSTUV-DA-89876; insbes. Kap. 13.2, S. 519; Kap. 13.5, S. 529</li></ul> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bose-Einstein-Statistik&veaction=edit&section=5" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bose-Einstein-Statistik&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fermi-Dirac-Statistik" title="Fermi-Dirac-Statistik">Fermi-Dirac-Statistik</a></li> <li><a href="/wiki/Boltzmann-Statistik" title="Boltzmann-Statistik">Boltzmann-Statistik</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bose-Einstein-Statistik&veaction=edit&section=6" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bose-Einstein-Statistik&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Commons"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" 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title="Betaverteilung">Beta</a> | <a href="/wiki/Cantor-Verteilung" title="Cantor-Verteilung">Cantor</a> | <a href="/w/index.php?title=Kumaraswamy-Verteilung&action=edit&redlink=1" class="new" title="Kumaraswamy-Verteilung (Seite nicht vorhanden)">Kumaraswamy</a> | <a href="/w/index.php?title=Raised-Cosine-Verteilung&action=edit&redlink=1" class="new" title="Raised-Cosine-Verteilung (Seite nicht vorhanden)">raised Cosine</a> | <a href="/wiki/Dreiecksverteilung" title="Dreiecksverteilung">Dreieck</a> | <a href="/wiki/Trapezverteilung" title="Trapezverteilung">Trapez</a> | <a href="/w/index.php?title=U-quadratische_Verteilung&action=edit&redlink=1" class="new" title="U-quadratische Verteilung (Seite nicht vorhanden)">U-quadratisch</a> | <a href="/wiki/Stetige_Gleichverteilung" title="Stetige Gleichverteilung">stetig uniform</a> | <a href="/w/index.php?title=Wigner-Halbkreis-Verteilung&action=edit&redlink=1" class="new" title="Wigner-Halbkreis-Verteilung (Seite nicht vorhanden)">Wigner-Halbkreis</a> </p><p><b>Kontinuierliche univariate Verteilungen mit halboffenem Intervall:</b><br /> <a href="/wiki/Beta-prime-Verteilung" class="mw-redirect" title="Beta-prime-Verteilung">Beta prime</a> | <a class="mw-selflink selflink">Bose-Einstein</a> | <a href="/w/index.php?title=Burr-Verteilung&action=edit&redlink=1" class="new" title="Burr-Verteilung (Seite nicht vorhanden)">Burr</a> | <a href="/wiki/Chi-Verteilung" title="Chi-Verteilung">Chi</a> | <a href="/wiki/Chi-Quadrat-Verteilung" title="Chi-Quadrat-Verteilung">Chi-Quadrat</a> | <a href="/w/index.php?title=Coxian-Verteilung&action=edit&redlink=1" class="new" title="Coxian-Verteilung (Seite nicht vorhanden)">Coxian</a> | <a href="/wiki/Erlang-Verteilung" title="Erlang-Verteilung">Erlang</a> | <a href="/wiki/Exponentialverteilung" title="Exponentialverteilung">Exponential</a> | <a href="/wiki/Extremwertverteilung" title="Extremwertverteilung">Extremwert</a> | <a href="/wiki/F-Verteilung" title="F-Verteilung">F</a> | <a href="/wiki/Fermi-Dirac-Statistik" title="Fermi-Dirac-Statistik">Fermi-Dirac</a> | <a href="/w/index.php?title=Folded_Normalverteilung&action=edit&redlink=1" class="new" title="Folded Normalverteilung (Seite nicht vorhanden)">Folded normal</a> | <a href="/wiki/Fr%C3%A9chet-Verteilung" title="Fréchet-Verteilung">Fréchet</a> | <a href="/wiki/Gammaverteilung" title="Gammaverteilung">Gamma</a> | <a href="/wiki/Gamma-Gamma-Verteilung" title="Gamma-Gamma-Verteilung">Gamma-Gamma</a> | <a href="/w/index.php?title=Verallgemeinerte_inverse_Gau%C3%9F%E2%80%99sche_Verteilung&action=edit&redlink=1" class="new" title="Verallgemeinerte inverse Gauß’sche Verteilung (Seite nicht vorhanden)">verallgemeinert invers Gauß</a> | <a href="/w/index.php?title=Halblogistische_Verteilung&action=edit&redlink=1" class="new" title="Halblogistische Verteilung (Seite nicht vorhanden)">halblogistisch</a> | <a href="/w/index.php?title=Halbnormale_Verteilung&action=edit&redlink=1" class="new" title="Halbnormale Verteilung (Seite nicht vorhanden)">halbnormal</a> | <a href="/wiki/Hartman-Watson-Verteilung" title="Hartman-Watson-Verteilung">Hartman-Watson</a> | <a href="/wiki/Hotellings_T-Quadrat-Verteilung" class="mw-redirect" title="Hotellings T-Quadrat-Verteilung">Hotellings T-Quadrat</a> | <a href="/wiki/Hyper-exponentiale_Verteilung" class="mw-redirect" title="Hyper-exponentiale Verteilung">hyper-exponentiale</a> | <a href="/w/index.php?title=Hypoexponentiale_Verteilung&action=edit&redlink=1" class="new" title="Hypoexponentiale Verteilung (Seite nicht vorhanden)">hypoexponential</a> | <a href="/w/index.php?title=Inverse_Chi-Quadrat-Verteilung&action=edit&redlink=1" class="new" title="Inverse Chi-Quadrat-Verteilung (Seite nicht vorhanden)">invers Chi-Quadrat</a> | <a href="/w/index.php?title=Scale-inverse_Chi-Quadrat-Verteilung&action=edit&redlink=1" class="new" title="Scale-inverse Chi-Quadrat-Verteilung (Seite nicht vorhanden)">scale-invers Chi-Quadrat</a> | <a href="/wiki/Inverse_Normalverteilung" title="Inverse Normalverteilung">Invers Normal</a> | <a href="/w/index.php?title=Inverse_Gamma-Verteilung&action=edit&redlink=1" class="new" title="Inverse Gamma-Verteilung (Seite nicht vorhanden)">Invers Gamma</a> | <a href="/wiki/Kolmogorow-Verteilung" title="Kolmogorow-Verteilung">Kolmogorow-Verteilung</a> | <a href="/wiki/L%C3%A9vy-Verteilung" title="Lévy-Verteilung">Lévy</a> | <a href="/wiki/Logarithmische_Normalverteilung" title="Logarithmische Normalverteilung">log-normal</a> | <a href="/w/index.php?title=Log-logistische_Verteilung&action=edit&redlink=1" class="new" title="Log-logistische Verteilung (Seite nicht vorhanden)">log-logistisch</a> | <a href="/wiki/Maxwell-Boltzmann-Verteilung" title="Maxwell-Boltzmann-Verteilung">Maxwell-Boltzmann</a> | <a href="/w/index.php?title=Maxwell-Speed-Verteilung&action=edit&redlink=1" class="new" title="Maxwell-Speed-Verteilung (Seite nicht vorhanden)">Maxwell-Speed</a> | <a href="/w/index.php?title=Nakagami-Verteilung&action=edit&redlink=1" class="new" title="Nakagami-Verteilung (Seite nicht vorhanden)">Nakagami</a> | <a href="/wiki/Nichtzentrierte_Chi-Quadrat-Verteilung" class="mw-redirect" title="Nichtzentrierte Chi-Quadrat-Verteilung">nichtzentriert Chi-Quadrat</a> | <a href="/wiki/Pareto-Verteilung" title="Pareto-Verteilung">Pareto</a> | <a href="/w/index.php?title=Phase-Type-Verteilung&action=edit&redlink=1" class="new" title="Phase-Type-Verteilung (Seite nicht vorhanden)">Phase-Type</a> | <a href="/wiki/Rayleigh-Verteilung" title="Rayleigh-Verteilung">Rayleigh</a> | <a href="/w/index.php?title=Relativistische_Breit-Wigner-Verteilung&action=edit&redlink=1" class="new" title="Relativistische Breit-Wigner-Verteilung (Seite nicht vorhanden)">relativistisch Breit-Wigner</a> | <a href="/w/index.php?title=Rice-Verteilung&action=edit&redlink=1" class="new" title="Rice-Verteilung (Seite nicht vorhanden)">Rice</a> | <a href="/wiki/Rosin-Rammler-Verteilung" class="mw-redirect" title="Rosin-Rammler-Verteilung">Rosin-Rammler</a> | <a href="/w/index.php?title=Shifted_Gompertz-Verteilung&action=edit&redlink=1" class="new" title="Shifted Gompertz-Verteilung (Seite nicht vorhanden)">shifted Gompertz</a> | <a href="/w/index.php?title=Truncated_Normalverteilung&action=edit&redlink=1" class="new" title="Truncated Normalverteilung (Seite nicht vorhanden)">truncated normal</a> | <a href="/w/index.php?title=Type-2-Gumbel-Verteilung&action=edit&redlink=1" class="new" title="Type-2-Gumbel-Verteilung (Seite nicht vorhanden)">Type-2-Gumbel</a> | <a href="/wiki/Weibull-Verteilung" title="Weibull-Verteilung">Weibull</a> | <a href="/w/index.php?title=Wilks%E2%80%99_Lambda-Verteilung&action=edit&redlink=1" class="new" title="Wilks’ Lambda-Verteilung (Seite nicht vorhanden)">Wilks’ Lambda</a> </p><p><b>Kontinuierliche univariate Verteilungen mit unbeschränktem Intervall:</b><br /> <a href="/wiki/Cauchy-Verteilung" title="Cauchy-Verteilung">Cauchy</a> | <a href="/wiki/Extremwertverteilung" title="Extremwertverteilung">Extremwert</a> | <a href="/w/index.php?title=Exponential-Power-Verteilung&action=edit&redlink=1" class="new" title="Exponential-Power-Verteilung (Seite nicht vorhanden)">exponential Power</a> | <a href="/wiki/Fishers_z-Verteilung" class="mw-redirect" title="Fishers z-Verteilung">Fishers <i>z</i></a> | <a href="/wiki/Gumbel-Verteilung" title="Gumbel-Verteilung">Fisher-Tippett (Gumbel)</a> | <a href="/w/index.php?title=Generalized-hyperbolic-Verteilung&action=edit&redlink=1" class="new" title="Generalized-hyperbolic-Verteilung (Seite nicht vorhanden)">generalized hyperbolic</a> | <a href="/w/index.php?title=Hyperbolic-secant-Verteilung&action=edit&redlink=1" class="new" title="Hyperbolic-secant-Verteilung (Seite nicht vorhanden)">Hyperbolic-secant</a> | <a href="/wiki/Landauverteilung" title="Landauverteilung">Landau</a> | <a href="/wiki/Laplace-Verteilung" title="Laplace-Verteilung">Laplace</a> | <a href="/wiki/Alpha-stabile_Verteilungen" title="Alpha-stabile Verteilungen">alpha-stabil</a> | <a href="/wiki/Logistische_Verteilung" title="Logistische Verteilung">logistisch</a> | <a href="/wiki/Normalverteilung" title="Normalverteilung">normal (Gauß)</a> | <a href="/w/index.php?title=Normal-inverse_Gau%C3%9F%E2%80%99sche_Verteilung&action=edit&redlink=1" class="new" title="Normal-inverse Gauß’sche Verteilung (Seite nicht vorhanden)">normal-invers Gauß’sch</a> | <a href="/w/index.php?title=Skew-normal-Verteilung&action=edit&redlink=1" class="new" title="Skew-normal-Verteilung (Seite nicht vorhanden)">Skew-normal</a> | <a href="/wiki/Studentsche_t-Verteilung" title="Studentsche t-Verteilung">Studentsche t</a> | <a href="/w/index.php?title=Type-1-Gumbel-Verteilung&action=edit&redlink=1" class="new" title="Type-1-Gumbel-Verteilung (Seite nicht vorhanden)">Type-1-Gumbel</a> | <a href="/w/index.php?title=Variance-Gamma-Verteilung&action=edit&redlink=1" class="new" 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بوز وأينشتاين – Arabisch" lang="ar" hreflang="ar" data-title="إحصاء بوز وأينشتاين" data-language-autonym="العربية" data-language-local-name="Arabisch" 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 – Assamesisch" lang="as" hreflang="as" data-title="বসু-আইনষ্টাইন পৰিসংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamesisch" 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 – Bulgarisch" lang="bg" hreflang="bg" data-title="Статистика на Бозе-Айнщайн" data-language-autonym="Български" data-language-local-name="Bulgarisch" 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 – Bengalisch" lang="bn" hreflang="bn" data-title="বসু-আইনস্টাইন পরিসংখ্যান" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" 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 – Bosnisch" lang="bs" hreflang="bs" data-title="Bose–Einsteinova statistika" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" 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 – Katalanisch" lang="ca" hreflang="ca" data-title="Estadística de Bose-Einstein" data-language-autonym="Català" data-language-local-name="Katalanisch" 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 – Tschechisch" lang="cs" hreflang="cs" data-title="Boseho–Einsteinovo rozdělení" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</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 – Englisch" lang="en" hreflang="en" data-title="Bose–Einstein statistics" data-language-autonym="English" data-language-local-name="Englisch" 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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estad%C3%ADstica_de_Bose-Einstein" title="Estadística de Bose-Einstein – Spanisch" lang="es" hreflang="es" data-title="Estadística de Bose-Einstein" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</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 – Estnisch" lang="et" hreflang="et" data-title="Bose-Einsteini statistika" data-language-autonym="Eesti" data-language-local-name="Estnisch" 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 – Baskisch" lang="eu" hreflang="eu" data-title="Bose-Einstein estatistika" data-language-autonym="Euskara" data-language-local-name="Baskisch" 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 – Persisch" lang="fa" hreflang="fa" data-title="آمار بوز-اینشتین" data-language-autonym="فارسی" data-language-local-name="Persisch" 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 – Finnisch" lang="fi" hreflang="fi" data-title="Bosen–Einsteinin statistiikka" data-language-autonym="Suomi" data-language-local-name="Finnisch" 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 – Französisch" lang="fr" hreflang="fr" data-title="Statistique de Bose-Einstein" data-language-autonym="Français" data-language-local-name="Französisch" 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 – Irisch" lang="ga" hreflang="ga" data-title="Staitistic Bose-Einstein" data-language-autonym="Gaeilge" data-language-local-name="Irisch" 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 – Galicisch" lang="gl" hreflang="gl" data-title="Estatística de Bose-Einstein" data-language-autonym="Galego" data-language-local-name="Galicisch" 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 – Gujarati" lang="gu" hreflang="gu" data-title="બોઝ-આઈન્સ્ટાઈન આંકડાશાસ્ત્ર" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" 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 – Hebräisch" lang="he" hreflang="he" data-title="התפלגות בוז-איינשטיין" data-language-autonym="עברית" data-language-local-name="Hebräisch" 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 – Kroatisch" lang="hr" hreflang="hr" data-title="Bose-Einsteinova statistika" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" 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 – Ungarisch" lang="hu" hreflang="hu" data-title="Bose–Einstein-statisztika" data-language-autonym="Magyar" data-language-local-name="Ungarisch" 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 – Armenisch" lang="hy" hreflang="hy" data-title="Բոզե-Այնշտայնի վիճակագրություն" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" 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 – Indonesisch" lang="id" hreflang="id" data-title="Statistik Bose–Einstein" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" 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 – Italienisch" lang="it" hreflang="it" data-title="Statistica di Bose-Einstein" data-language-autonym="Italiano" data-language-local-name="Italienisch" 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 – Japanisch" lang="ja" hreflang="ja" data-title="ボース分布関数" data-language-autonym="日本語" data-language-local-name="Japanisch" 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 – Georgisch" lang="ka" hreflang="ka" data-title="ბოზე-აინშტაინის სტატისტიკა" data-language-autonym="ქართული" data-language-local-name="Georgisch" 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 – Kasachisch" lang="kk" hreflang="kk" data-title="Бозе–Эйнштейн статистикасы" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" 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 – Kannada" lang="kn" hreflang="kn" data-title="ಬೋಸ್-ಐನ್‍ಸ್ಟೈನ್ ಸಂಖ್ಯಾಕಲನವಿಜ್ಞಾನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" 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 – Koreanisch" lang="ko" hreflang="ko" data-title="보스-아인슈타인 통계" data-language-autonym="한국어" data-language-local-name="Koreanisch" 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 – Niederländisch" lang="nl" hreflang="nl" data-title="Bose-Einsteinstatistiek" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" 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 – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Bose–Einstein-statistikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (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 – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Bose–Einstein-statistikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" 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 – Polnisch" lang="pl" hreflang="pl" data-title="Statystyka Bosego-Einsteina" data-language-autonym="Polski" data-language-local-name="Polnisch" 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 – Portugiesisch" lang="pt" hreflang="pt" data-title="Estatística de Bose-Einstein" data-language-autonym="Português" data-language-local-name="Portugiesisch" 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 – Rumänisch" lang="ro" hreflang="ro" data-title="Statistica Bose-Einstein" data-language-autonym="Română" data-language-local-name="Rumänisch" 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 – Russisch" lang="ru" hreflang="ru" data-title="Статистика Бозе — Эйнштейна" data-language-autonym="Русский" data-language-local-name="Russisch" 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 – Serbokroatisch" lang="sh" hreflang="sh" data-title="Bose-Einsteinova statistika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" 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 – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Bose-Einstein statistics" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" 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 – Slowakisch" lang="sk" hreflang="sk" data-title="Boseho-Einsteinovo rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" 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 – Slowenisch" lang="sl" hreflang="sl" data-title="Bose-Einsteinova statistika" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" 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 – Schwedisch" lang="sv" hreflang="sv" data-title="Bose–Einstein-statistik" data-language-autonym="Svenska" data-language-local-name="Schwedisch" 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 – Tatarisch" lang="tt" hreflang="tt" data-title="Бозе — Эйнштейн бүленеше" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatarisch" 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 – Ukrainisch" lang="uk" hreflang="uk" data-title="Статистика Бозе — Ейнштейна" data-language-autonym="Українська" data-language-local-name="Ukrainisch" 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 – Usbekisch" lang="uz" hreflang="uz" data-title="Boze Eynshteyn statistikasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Th%E1%BB%91ng_k%C3%AA_Bose%E2%80%93Einstein" title="Thống kê Bose–Einstein – Vietnamesisch" lang="vi" hreflang="vi" data-title="Thống kê Bose–Einstein" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/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 – Chinesisch" lang="zh" hreflang="zh" data-title="玻色–爱因斯坦统计" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q191076#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 4. 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Bose-Einstein-Statistik – Wikipedia