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Das ideale Bosegas Eq: math.2561.0

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TeX (original user input):

\begin{align}
  & Y=\sum\limits_{{{N}_{1}}...{{N}_{l}}=0}^{\infty }{{}}\exp \left( -\beta \sum\limits_{j=1}^{l}{{}}\left( {{N}_{j}}{{E}_{j}}-\mu {{N}_{j}} \right) \right)=\prod\limits_{j=1}^{l}{{}}\left( \sum\limits_{{{N}_{j}}=0}^{\infty }{{}}\exp \left( -\beta \left( {{N}_{j}}{{E}_{j}}-\mu {{N}_{j}} \right) \right) \right) \\ 
 & =\prod\limits_{j=1}^{l}{{}}\left( \sum\limits_{{{N}_{j}}=0}^{\infty }{{}}{{t}_{j}}^{{{N}_{j}}} \right) \\ 
 & {{t}_{j}}:=\exp \left( -\beta \left( {{E}_{j}}-\mu  \right) \right) \\ 
 & Y=\prod\limits_{j=1}^{l}{{}}\frac{1}{1-{{t}_{j}}}=\prod\limits_{j=1}^{l}{{}}{{Y}_{j}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&Y=\sum \limits _{{{N}_{1}}...{{N}_{l}}=0}^{\infty }{}\exp \left(-\beta \sum \limits _{j=1}^{l}{}\left({{N}_{j}}{{E}_{j}}-\mu {{N}_{j}}\right)\right)=\prod \limits _{j=1}^{l}{}\left(\sum \limits _{{{N}_{j}}=0}^{\infty }{}\exp \left(-\beta \left({{N}_{j}}{{E}_{j}}-\mu {{N}_{j}}\right)\right)\right)\\&=\prod \limits _{j=1}^{l}{}\left(\sum \limits _{{{N}_{j}}=0}^{\infty }{}{{t}_{j}}^{{N}_{j}}\right)\\&{{t}_{j}}:=\exp \left(-\beta \left({{E}_{j}}-\mu \right)\right)\\&Y=\prod \limits _{j=1}^{l}{}{\frac {1}{1-{{t}_{j}}}}=\prod \limits _{j=1}^{l}{}{{Y}_{j}}\\\end{aligned}}

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\begin{aligned} Y = \sum_{N_{1} . . . N_{l} = 0}^{\infty} \exp (- β \sum_{j = 1}^{l} (N_{j} E_{j} - μ N_{j})) = \prod_{j = 1}^{l} (\sum_{N_{j} = 0}^{\infty} \exp (- β (N_{j} E_{j} - μ N_{j}))) \\ = \prod_{j = 1}^{l} (\sum_{N_{j} = 0}^{\infty} {t_{j}}^{N_{j}}) \\ t_{j} : = \exp (- β (E_{j} - μ)) \\ Y = \prod_{j = 1}^{l} \frac{1}{1 - t_{j}} = \prod_{j = 1}^{l} Y_{j} \end{aligned}

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Identifiers

$Y$

$N_{1}$

$N_{l}$

$β$

$j$

$l$

$N_{j}$

$E_{j}$

$μ$

$N_{j}$

$j$

$l$

$N_{j}$

$β$

$N_{j}$

$E_{j}$

$μ$

$N_{j}$

$j$

$l$

$N_{j}$

$t_{j}$

$N_{j}$

$t_{j}$

$β$

$E_{j}$

$μ$

$Y$

$j$

$l$

$t_{j}$

$j$

$l$

$Y_{j}$

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