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Das ideale Bosegas Eq: math.2561.5

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TeX (original user input):

\begin{align}
  & \left\langle {{N}_{j}} \right\rangle =\sum\limits_{{{N}_{j}}=0}^{\infty }{{}}{{N}_{j}}p({{N}_{j}})=\sum\limits_{{{N}_{j}}=0}^{\infty }{{}}{{N}_{j}}\left( 1-{{t}_{j}} \right){{t}_{j}}^{{{N}_{j}}}=\left( 1-{{t}_{j}} \right){{t}_{j}}\frac{d}{d{{t}_{j}}}\sum\limits_{{{N}_{j}}=0}^{\infty }{{}}{{t}_{j}}^{{{N}_{j}}} \\ 
 & =\left( 1-{{t}_{j}} \right){{t}_{j}}\frac{d}{d{{t}_{j}}}\left( \frac{1}{1-{{t}_{j}}} \right)=\left( 1-{{t}_{j}} \right){{t}_{j}}\left( \frac{1}{{{\left( 1-{{t}_{j}} \right)}^{2}}} \right)=\frac{{{t}_{j}}}{\left( 1-{{t}_{j}} \right)} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&\left\langle {{N}_{j}}\right\rangle =\sum \limits _{{{N}_{j}}=0}^{\infty }{}{{N}_{j}}p({{N}_{j}})=\sum \limits _{{{N}_{j}}=0}^{\infty }{}{{N}_{j}}\left(1-{{t}_{j}}\right){{t}_{j}}^{{N}_{j}}=\left(1-{{t}_{j}}\right){{t}_{j}}{\frac {d}{d{{t}_{j}}}}\sum \limits _{{{N}_{j}}=0}^{\infty }{}{{t}_{j}}^{{N}_{j}}\\&=\left(1-{{t}_{j}}\right){{t}_{j}}{\frac {d}{d{{t}_{j}}}}\left({\frac {1}{1-{{t}_{j}}}}\right)=\left(1-{{t}_{j}}\right){{t}_{j}}\left({\frac {1}{{\left(1-{{t}_{j}}\right)}^{2}}}\right)={\frac {{t}_{j}}{\left(1-{{t}_{j}}\right)}}\\\end{aligned}}

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\begin{aligned} ⟨ N_{j} ⟩ = \sum_{N_{j} = 0}^{\infty} N_{j} p (N_{j}) = \sum_{N_{j} = 0}^{\infty} N_{j} (1 - t_{j}) {t_{j}}^{N_{j}} = (1 - t_{j}) t_{j} \frac{d}{d t_{j}} \sum_{N_{j} = 0}^{\infty} {t_{j}}^{N_{j}} \\ = (1 - t_{j}) t_{j} \frac{d}{d t_{j}} (\frac{1}{1 - t_{j}}) = (1 - t_{j}) t_{j} (\frac{1}{{(1 - t_{j})}^{2}}) = \frac{t_{j}}{(1 - t_{j})} \end{aligned}

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Identifiers

$N_{j}$

$N_{j}$

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$p$

$N_{j}$

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$t_{j}$

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$N_{j}$

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$d$

$d$

$t_{j}$

$N_{j}$

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$N_{j}$

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$d$

$d$

$t_{j}$

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