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Display information for equation id:math.2273.97 on revision:2273

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Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt Eq: math.2273.97

Hash: b6e1d48bc5018c8a32a0fe03941f25b8

TeX (original user input):

\begin{align}
  & 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 }}} \\
 & =k\sum\limits_{\alpha }{\operatorname{Tr}\left( -\sum\limits_{\nu }{{{\lambda }_{\nu }}}\frac{\partial {{G}_{\nu }}}{\partial {{h}_{\alpha }}}R \right)d{{h}_{\alpha }}} \\
 & =-k\sum\limits_{\alpha ,\nu }{{{\lambda }_{\nu }}\left\langle \frac{\partial {{G}_{\nu }}}{\partial {{h}_{\alpha }}} \right\rangle d{{h}_{\alpha }}} 
\end{align}

TeX (checked):

{\begin{aligned}&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 }}}\\&=k\sum \limits _{\alpha }{\operatorname {Tr} \left(-\sum \limits _{\nu }{{\lambda }_{\nu }}{\frac {\partial {{G}_{\nu }}}{\partial {{h}_{\alpha }}}}R\right)d{{h}_{\alpha }}}\\&=-k\sum \limits _{\alpha ,\nu }{{{\lambda }_{\nu }}\left\langle {\frac {\partial {{G}_{\nu }}}{\partial {{h}_{\alpha }}}}\right\rangle d{{h}_{\alpha }}}\end{aligned}}

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\begin{aligned} 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} \\ = k \sum_{α} Tr (- \sum_{ν} λ_{ν} \frac{\partial G_{ν}}{\partial h_{α}} R) d h_{α} \\ = - k \sum_{α, ν} λ_{ν} ⟨ \frac{\partial G_{ν}}{\partial h_{α}} ⟩ d h_{α} \end{aligned}

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Identifiers

$k$

$α$

$Z$

$Z$

$h_{α}$

$d$

$h_{α}$

$k$

$α$

$Z$

$h_{α}$

$ν$

$λ_{ν}$

$G_{ν}$

$d$

$h_{α}$

$k$

$α$

$ν$

$λ_{ν}$

$G_{ν}$

$h_{α}$

$R$

$d$

$h_{α}$

$k$

$α$

$ν$

$λ_{ν}$

$G_{ν}$

$h_{α}$

$d$

$h_{α}$

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