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Display information for equation id:math.2270.96 on revision:2270

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Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt Eq: math.2273.96

Hash: 9edc39ba546455818572ed237f6a8d87

TeX (original user input):

\begin{align}
  & S=k\sum\limits_{\nu }{{{\lambda }_{\nu }}\left\langle {{G}_{\nu }} \right\rangle }+k\ln Z \\
 & dS=k\sum\limits_{\nu }{\left( d{{\lambda }_{\nu }}\left\langle {{G}_{\nu }} \right\rangle +{{\lambda }_{\nu }}d\left\langle {{G}_{\nu }} \right\rangle  \right)}+k\frac{dZ}{Z} \\
 & =k\underbrace{\sum\limits_{\nu }{{{\lambda }_{\nu }}d\left\langle {{G}_{\nu }} \right\rangle }}_{\begin{smallmatrix}
 \text{Teil der} \\
 \text{Gibbsgleichung}
\end{smallmatrix}}+k\sum\limits_{\alpha }{\frac{1}{Z}\frac{\partial Z}{\partial {{h}_{\alpha }}}d{{h}_{\alpha }}} \\
 & k\sum\limits_{\alpha }{\frac{1}{Z}\frac{\partial Z}{\partial {{h}_{\alpha }}}d{{h}_{\alpha }}}=k\sum\limits_{\alpha }{\frac{1}{Z}\operatorname{Tr}\left( \frac{\partial }{\partial {{h}_{\alpha }}}{{\operatorname{e}}^{-\sum\limits_{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}} \right)d{{h}_{\alpha }}} 
\end{align}

TeX (checked):

{\begin{aligned}&S=k\sum \limits _{\nu }{{{\lambda }_{\nu }}\left\langle {{G}_{\nu }}\right\rangle }+k\ln Z\\&dS=k\sum \limits _{\nu }{\left(d{{\lambda }_{\nu }}\left\langle {{G}_{\nu }}\right\rangle +{{\lambda }_{\nu }}d\left\langle {{G}_{\nu }}\right\rangle \right)}+k{\frac {dZ}{Z}}\\&=k\underbrace {\sum \limits _{\nu }{{{\lambda }_{\nu }}d\left\langle {{G}_{\nu }}\right\rangle }} _{\begin{smallmatrix}{\text{Teil der}}\\{\text{Gibbsgleichung}}\end{smallmatrix}}+k\sum \limits _{\alpha }{{\frac {1}{Z}}{\frac {\partial Z}{\partial {{h}_{\alpha }}}}d{{h}_{\alpha }}}\\&k\sum \limits _{\alpha }{{\frac {1}{Z}}{\frac {\partial Z}{\partial {{h}_{\alpha }}}}d{{h}_{\alpha }}}=k\sum \limits _{\alpha }{{\frac {1}{Z}}\operatorname {Tr} \left({\frac {\partial }{\partial {{h}_{\alpha }}}}{{\operatorname {e} }^{-\sum \limits _{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}}\right)d{{h}_{\alpha }}}\end{aligned}}

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\begin{aligned} S = k \sum_{ν} λ_{ν} ⟨ G_{ν} ⟩ + k \ln Z \\ d S = k \sum_{ν} (d λ_{ν} ⟨ G_{ν} ⟩ + λ_{ν} d ⟨ G_{ν} ⟩) + k \frac{d Z}{Z} \\ = k \underset{\begin{matrix} Teil der \\ Gibbsgleichung \end{matrix}}{\underset{⏟}{\sum_{ν} λ_{ν} d ⟨ G_{ν} ⟩}} + k \sum_{α} \frac{1}{Z} \frac{\partial Z}{\partial h_{α}} d h_{α} \\ k \sum_{α} \frac{1}{Z} \frac{\partial Z}{\partial h_{α}} d h_{α} = k \sum_{α} \frac{1}{Z} Tr (\frac{\partial}{\partial h_{α}} e^{- \sum_{ν} λ_{ν} G_{ν}}) d h_{α} \end{aligned}

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Identifiers

$S$

$k$

$ν$

$λ_{ν}$

$G_{ν}$

$k$

$Z$

$d$

$S$

$k$

$ν$

$d$

$λ_{ν}$

$G_{ν}$

$λ_{ν}$

$d$

$G_{ν}$

$k$

$d$

$Z$

$Z$

$k$

$ν$

$λ_{ν}$

$d$

$G_{ν}$

$k$

$α$

$Z$

$Z$

$h_{α}$

$d$

$h_{α}$

$k$

$α$

$Z$

$Z$

$h_{α}$

$d$

$h_{α}$

$k$

$α$

$Z$

$h_{α}$

$ν$

$λ_{ν}$

$G_{ν}$

$d$

$h_{α}$

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