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Display information for equation id:math.940.55 on revision:940

* Page found: Verallgemeinerte kanonische Verteilung (eq math.940.55)

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Verallgemeinerte kanonische Verteilung Eq: math.940.55

Hash: 090e2914c4c302daa71349ee9ee5e3dc

TeX (original user input):

\begin{align}
  & \Psi \left( {{\lambda }_{1}},...{{\lambda }_{m}} \right)=-\ln \sum\limits_{i}^{{}}{{}}\exp \left( -{{\lambda }_{n}}{{M}_{i}}^{n} \right) \\ 
 & \Rightarrow \frac{\partial }{\partial {{\lambda }_{n}}}\Psi =-\frac{\sum\limits_{i}^{{}}{{}}\left( -{{M}_{i}}^{n} \right)\exp \left( -{{\lambda }_{n}}{{M}_{i}}^{n} \right)}{\sum\limits_{i}^{{}}{{}}\exp \left( -{{\lambda }_{n}}{{M}_{i}}^{n} \right)} \\  
 & \sum\limits_{i}^{{}}{{}}\exp \left( -{{\lambda }_{n}}{{M}_{i}}^{n} \right)={{e}^{-\Psi }} \\ 
 & \Rightarrow \frac{\partial }{\partial {{\lambda }_{n}}}\Psi =\sum\limits_{i}^{{}}{{}}\left( {{M}_{i}}^{n} \right)\exp \left( \Psi -{{\lambda }_{n}}{{M}_{i}}^{n} \right) \\ 
 & \exp \left( \Psi -{{\lambda }_{n}}{{M}_{i}}^{n} \right)={{P}_{i}} \\ 
 & \Rightarrow \frac{\partial }{\partial {{\lambda }_{n}}}\Psi =\sum\limits_{i}^{{}}{{}}\left( {{M}_{i}}^{n} \right){{P}_{i}} \\ 
 & \Rightarrow \frac{\partial }{\partial {{\lambda }_{n}}}\Psi =\left\langle {{M}^{n}} \right\rangle  \\ 
\end{align}

TeX (checked):

{\begin{aligned}&\Psi \left({{\lambda }_{1}},...{{\lambda }_{m}}\right)=-\ln \sum \limits _{i}^{}{}\exp \left(-{{\lambda }_{n}}{{M}_{i}}^{n}\right)\\&\Rightarrow {\frac {\partial }{\partial {{\lambda }_{n}}}}\Psi =-{\frac {\sum \limits _{i}^{}{}\left(-{{M}_{i}}^{n}\right)\exp \left(-{{\lambda }_{n}}{{M}_{i}}^{n}\right)}{\sum \limits _{i}^{}{}\exp \left(-{{\lambda }_{n}}{{M}_{i}}^{n}\right)}}\\&\sum \limits _{i}^{}{}\exp \left(-{{\lambda }_{n}}{{M}_{i}}^{n}\right)={{e}^{-\Psi }}\\&\Rightarrow {\frac {\partial }{\partial {{\lambda }_{n}}}}\Psi =\sum \limits _{i}^{}{}\left({{M}_{i}}^{n}\right)\exp \left(\Psi -{{\lambda }_{n}}{{M}_{i}}^{n}\right)\\&\exp \left(\Psi -{{\lambda }_{n}}{{M}_{i}}^{n}\right)={{P}_{i}}\\&\Rightarrow {\frac {\partial }{\partial {{\lambda }_{n}}}}\Psi =\sum \limits _{i}^{}{}\left({{M}_{i}}^{n}\right){{P}_{i}}\\&\Rightarrow {\frac {\partial }{\partial {{\lambda }_{n}}}}\Psi =\left\langle {{M}^{n}}\right\rangle \\\end{aligned}}

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\begin{aligned} Ψ (λ_{1}, . . . λ_{m}) = - \ln \sum_{i}^{} \exp (- λ_{n} {M_{i}}^{n}) \\ \Rightarrow \frac{\partial}{\partial λ_{n}} Ψ = - \frac{\sum_{i}^{} (- {M_{i}}^{n}) \exp (- λ_{n} {M_{i}}^{n})}{\sum_{i}^{} \exp (- λ_{n} {M_{i}}^{n})} \\ \sum_{i}^{} \exp (- λ_{n} {M_{i}}^{n}) = e^{- Ψ} \\ \Rightarrow \frac{\partial}{\partial λ_{n}} Ψ = \sum_{i}^{} ({M_{i}}^{n}) \exp (Ψ - λ_{n} {M_{i}}^{n}) \\ \exp (Ψ - λ_{n} {M_{i}}^{n}) = P_{i} \\ \Rightarrow \frac{\partial}{\partial λ_{n}} Ψ = \sum_{i}^{} ({M_{i}}^{n}) P_{i} \\ \Rightarrow \frac{\partial}{\partial λ_{n}} Ψ = ⟨ M^{n} ⟩ \end{aligned}

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Identifiers

$Ψ$

$λ_{1}$

$λ_{m}$

$i$

$λ_{n}$

$M_{i}$

$n$

$λ_{n}$

$Ψ$

$i$

$M_{i}$

$n$

$λ_{n}$

$M_{i}$

$n$

$i$

$λ_{n}$

$M_{i}$

$n$

$i$

$λ_{n}$

$M_{i}$

$n$

$e$

$Ψ$

$λ_{n}$

$Ψ$

$i$

$M_{i}$

$n$

$Ψ$

$λ_{n}$

$M_{i}$

$n$

$Ψ$

$λ_{n}$

$M_{i}$

$n$

$P_{i}$

$λ_{n}$

$Ψ$

$i$

$M_{i}$

$n$

$P_{i}$

$λ_{n}$

$Ψ$

$M$

$n$

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