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Display information for equation id:math.929.99 on revision:929

* Page found: Verallgemeinerte kanonische Verteilung (eq math.929.99)

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Verallgemeinerte kanonische Verteilung Eq: math.940.99

Hash: 88682300bea1fce8a7a9bbe0132e7ff7

TeX (original user input):

\begin{align}
  & K\left( P,{{P}^{0}} \right)=\sum\limits_{i}^{{}}{{}}{{P}_{i}}\ln {{P}_{i}}-{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}-{{P}_{i}}^{0}\ln {{P}_{i}}^{0} \\ 
 & \sum\limits_{i}^{{}}{{}}{{P}_{i}}\ln {{P}_{i}}=I(P) \\ 
 & \sum\limits_{i}^{{}}{{}}{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=I({{P}^{0}}) \\ 
 & -{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=-\sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}-{{P}_{i}}^{0} \right)\ln {{P}_{i}}^{0} \\ 
 & \ln {{P}_{i}}^{0}=\Psi -{{\lambda }_{n}}{{M}_{i}}^{n} \\ 
 & -\sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}-{{P}_{i}}^{0} \right)\left( \Psi -{{\lambda }_{n}}{{M}_{i}}^{n} \right)={{\lambda }_{n}}\left( \sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}{{M}_{i}}^{n} \right)-\sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}^{0}{{M}_{i}}^{n} \right) \right) \\ 
 & \sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}{{M}_{i}}^{n} \right)=\left\langle {{M}^{n}} \right\rangle  \\ 
 & \sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}^{0}{{M}_{i}}^{n} \right)={{\left\langle {{M}^{n}} \right\rangle }_{0}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&K\left(P,{{P}^{0}}\right)=\sum \limits _{i}^{}{}{{P}_{i}}\ln {{P}_{i}}-{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}-{{P}_{i}}^{0}\ln {{P}_{i}}^{0}\\&\sum \limits _{i}^{}{}{{P}_{i}}\ln {{P}_{i}}=I(P)\\&\sum \limits _{i}^{}{}{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=I({{P}^{0}})\\&-{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=-\sum \limits _{i}^{}{}\left({{P}_{i}}-{{P}_{i}}^{0}\right)\ln {{P}_{i}}^{0}\\&\ln {{P}_{i}}^{0}=\Psi -{{\lambda }_{n}}{{M}_{i}}^{n}\\&-\sum \limits _{i}^{}{}\left({{P}_{i}}-{{P}_{i}}^{0}\right)\left(\Psi -{{\lambda }_{n}}{{M}_{i}}^{n}\right)={{\lambda }_{n}}\left(\sum \limits _{i}^{}{}\left({{P}_{i}}{{M}_{i}}^{n}\right)-\sum \limits _{i}^{}{}\left({{P}_{i}}^{0}{{M}_{i}}^{n}\right)\right)\\&\sum \limits _{i}^{}{}\left({{P}_{i}}{{M}_{i}}^{n}\right)=\left\langle {{M}^{n}}\right\rangle \\&\sum \limits _{i}^{}{}\left({{P}_{i}}^{0}{{M}_{i}}^{n}\right)={{\left\langle {{M}^{n}}\right\rangle }_{0}}\\\end{aligned}}

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\begin{aligned} K (P, P^{0}) = \sum_{i}^{} P_{i} \ln P_{i} - P_{i} \ln {P_{i}}^{0} + {P_{i}}^{0} \ln {P_{i}}^{0} - {P_{i}}^{0} \ln {P_{i}}^{0} \\ \sum_{i}^{} P_{i} \ln P_{i} = I (P) \\ \sum_{i}^{} {P_{i}}^{0} \ln {P_{i}}^{0} = I (P^{0}) \\ - P_{i} \ln {P_{i}}^{0} + {P_{i}}^{0} \ln {P_{i}}^{0} = - \sum_{i}^{} (P_{i} - {P_{i}}^{0}) \ln {P_{i}}^{0} \\ \ln {P_{i}}^{0} = Ψ - λ_{n} {M_{i}}^{n} \\ - \sum_{i}^{} (P_{i} - {P_{i}}^{0}) (Ψ - λ_{n} {M_{i}}^{n}) = λ_{n} (\sum_{i}^{} (P_{i} {M_{i}}^{n}) - \sum_{i}^{} ({P_{i}}^{0} {M_{i}}^{n})) \\ \sum_{i}^{} (P_{i} {M_{i}}^{n}) = ⟨ M^{n} ⟩ \\ \sum_{i}^{} ({P_{i}}^{0} {M_{i}}^{n}) = {⟨ M^{n} ⟩}_{0} \end{aligned}

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Identifiers

$K$

$P$

$P$

$i$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$i$

$P_{i}$

$P_{i}$

$I$

$P$

$i$

$P_{i}$

$P_{i}$

$I$

$P$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$i$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$Ψ$

$λ_{n}$

$M_{i}$

$n$

$i$

$P_{i}$

$P_{i}$

$Ψ$

$λ_{n}$

$M_{i}$

$n$

$λ_{n}$

$i$

$P_{i}$

$M_{i}$

$n$

$i$

$P_{i}$

$M_{i}$

$n$

$i$

$P_{i}$

$M_{i}$

$n$

$M$

$n$

$i$

$P_{i}$

$M_{i}$

$n$

$M$

$n$

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