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Das ideale Fermigas Eq: math.2555.29

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TeX (original user input):

\begin{align}

& \ln Y\approx \left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}{{p}^{2}}dp\ln \left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right) \\

& =\left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \left[ \left. \left( \frac{{{p}^{3}}}{3}\ln \left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right) \right) \right|_{0}^{\infty }-\int_{0}^{\infty }{{}}{{\frac{p}{3}}^{3}}\frac{-\beta \frac{p}{m}\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}}}{\left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right)}dp \right] \\

& \left. \left( \frac{{{p}^{3}}}{3}\ln \left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right) \right) \right|_{0}^{\infty }=0 \\

& \Rightarrow \ln Y=-\left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}{{\frac{p}{3}}^{3}}\frac{-\beta \frac{p}{m}\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}}}{\left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right)}dp=\frac{2}{3}\left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}dp{{p}^{2}}\frac{\beta \frac{{{p}^{2}}}{2m}}{\left( \frac{1}{\xi }{{e}^{\beta \frac{{{p}^{2}}}{2m}}}+1 \right)} \\

& =\frac{2}{3}\beta \left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}dp{{p}^{2}}\left\langle N(p) \right\rangle \frac{{{p}^{2}}}{2m} \\

\end{align}

TeX (checked):

{\begin{aligned}&\ln Y\approx \left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}{{p}^{2}}dp\ln \left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)\\&=\left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \left[\left.\left({\frac {{p}^{3}}{3}}\ln \left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)\right)\right|_{0}^{\infty }-\int _{0}^{\infty }{}{{\frac {p}{3}}^{3}}{\frac {-\beta {\frac {p}{m}}\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}}{\left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)}}dp\right]\\&\left.\left({\frac {{p}^{3}}{3}}\ln \left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)\right)\right|_{0}^{\infty }=0\\&\Rightarrow \ln Y=-\left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}{{\frac {p}{3}}^{3}}{\frac {-\beta {\frac {p}{m}}\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}}{\left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)}}dp={\frac {2}{3}}\left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}dp{{p}^{2}}{\frac {\beta {\frac {{p}^{2}}{2m}}}{\left({\frac {1}{\xi }}{{e}^{\beta {\frac {{p}^{2}}{2m}}}}+1\right)}}\\&={\frac {2}{3}}\beta \left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}dp{{p}^{2}}\left\langle N(p)\right\rangle {\frac {{p}^{2}}{2m}}\\\end{aligned}}

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\begin{aligned} \ln Y \approx (2 s + 1) (\frac{V}{h^{3}}) 4 π \int_{0}^{\infty} p^{2} d p \ln (1 + ξ e^{- β \frac{p^{2}}{2 m}}) \\ = (2 s + 1) (\frac{V}{h^{3}}) 4 π [{(\frac{p^{3}}{3} \ln (1 + ξ e^{- β \frac{p^{2}}{2 m}})) |}_{0}^{\infty} - \int_{0}^{\infty} {\frac{p}{3}}^{3} \frac{- β \frac{p}{m} ξ e^{- β \frac{p^{2}}{2 m}}}{(1 + ξ e^{- β \frac{p^{2}}{2 m}})} d p] \\ {(\frac{p^{3}}{3} \ln (1 + ξ e^{- β \frac{p^{2}}{2 m}})) |}_{0}^{\infty} = 0 \\ \Rightarrow \ln Y = - (2 s + 1) (\frac{V}{h^{3}}) 4 π \int_{0}^{\infty} {\frac{p}{3}}^{3} \frac{- β \frac{p}{m} ξ e^{- β \frac{p^{2}}{2 m}}}{(1 + ξ e^{- β \frac{p^{2}}{2 m}})} d p = \frac{2}{3} (2 s + 1) (\frac{V}{h^{3}}) 4 π \int_{0}^{\infty} d p p^{2} \frac{β \frac{p^{2}}{2 m}}{(\frac{1}{ξ} e^{β \frac{p^{2}}{2 m}} + 1)} \\ = \frac{2}{3} β (2 s + 1) (\frac{V}{h^{3}}) 4 π \int_{0}^{\infty} d p p^{2} ⟨ N (p) ⟩ \frac{p^{2}}{2 m} \end{aligned}

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$Y$

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$p$

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$ξ$

$e$

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$p$

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$π$

$p$

$p$

$N$

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