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Display information for equation id:math.2560.4 on revision:2560
* Page found: Das ideale Bosegas (eq math.2560.4)
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Hash: 1aac395f9822952cb5ac34fa8ca315c2
TeX (original user input):
\begin{align}
& \left\langle {{N}_{j}} \right\rangle =\frac{\partial {{\Psi }_{j}}}{\partial \alpha }=\frac{1}{\beta }\frac{\partial }{\partial \mu }\ln {{Y}_{j}}=-\frac{1}{\beta }\frac{\partial }{\partial \mu }\ln \left( 1-{{t}_{j}} \right)=\frac{{{t}_{j}}}{1-{{t}_{j}}}=\frac{1}{{{t}_{j}}^{-1}-1} \\
& \left\langle {{N}_{j}} \right\rangle =\frac{1}{\exp \left( \beta \left( {{E}_{j}}-\mu \right) \right)-1}=\frac{1}{\exp \left( \frac{\left( {{E}_{j}}-\mu \right)}{kT} \right)-1} \\
\end{align}
TeX (checked):
{\begin{aligned}&\left\langle {{N}_{j}}\right\rangle ={\frac {\partial {{\Psi }_{j}}}{\partial \alpha }}={\frac {1}{\beta }}{\frac {\partial }{\partial \mu }}\ln {{Y}_{j}}=-{\frac {1}{\beta }}{\frac {\partial }{\partial \mu }}\ln \left(1-{{t}_{j}}\right)={\frac {{t}_{j}}{1-{{t}_{j}}}}={\frac {1}{{{t}_{j}}^{-1}-1}}\\&\left\langle {{N}_{j}}\right\rangle ={\frac {1}{\exp \left(\beta \left({{E}_{j}}-\mu \right)\right)-1}}={\frac {1}{\exp \left({\frac {\left({{E}_{j}}-\mu \right)}{kT}}\right)-1}}\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>N</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi mathvariant="normal">Ψ</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>α</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>β</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>μ</mi></mrow></mrow></mfrac></mrow><mi>ln</mi><mo>⁡</mo><msub><mi>Y</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>β</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>μ</mi></mrow></mrow></mfrac></mrow><mi>ln</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>1</mn><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></mrow></msup><mo>−</mo><mn>1</mn></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>N</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo>−</mo><mi>μ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>1</mn></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo>−</mo><mi>μ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>1</mn></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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