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* Page found: Verallgemeinerte kanonische Verteilung (eq math.940.88)

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Verallgemeinerte kanonische Verteilung Eq: math.940.88

Hash: 2d6de575fe4034a2d3f8f7aeb1dcfb5e

TeX (original user input):

\begin{align}
  & \Gamma \left( \alpha  \right)=\ln \left\langle \exp \left( {{\alpha }_{n}}{{M}^{n}} \right) \right\rangle =\ln \sum\limits_{i}^{{}}{{}}{{P}_{i}}\exp \left( {{\alpha }_{n}}{{M}_{i}}^{n} \right)=\ln \sum\limits_{i}^{{}}{{}}{{e}^{\Psi -\left( {{\lambda }_{n}}-{{\alpha }_{n}} \right){{M}_{i}}^{n}}} \\ 
 & =\ln \left[ {{e}^{\Psi }}\cdot \sum\limits_{i}^{{}}{{}}{{e}^{-\left( {{\lambda }_{n}}-{{\alpha }_{n}} \right){{M}_{i}}^{n}}} \right]=\Psi \left( \lambda  \right)+\ln \left[ \sum\limits_{i}^{{}}{{}}{{e}^{-\left( {{\lambda }_{n}}-{{\alpha }_{n}} \right){{M}_{i}}^{n}}} \right] \\ 
 & \ln \left[ \sum\limits_{i}^{{}}{{}}{{e}^{-\left( {{\lambda }_{n}}-{{\alpha }_{n}} \right){{M}_{i}}^{n}}} \right]=-\Psi \left( \lambda -\alpha  \right) \\ 
 & \Rightarrow \Gamma \left( \alpha  \right)=\Psi \left( \lambda  \right)-\Psi \left( \lambda -\alpha  \right) \\ 
 & \Rightarrow {{Q}^{mn}}=-{{\left. \frac{{{\partial }^{2}}\Psi \left( \lambda -\alpha  \right)}{\partial {{\alpha }_{m}}\partial {{\alpha }_{n}}} \right|}_{\alpha =0}}=-\frac{{{\partial }^{2}}\Psi \left( \lambda  \right)}{\partial {{\lambda }_{m}}\partial {{\lambda }_{n}}}=-{{\eta }^{mn}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&\Gamma \left(\alpha \right)=\ln \left\langle \exp \left({{\alpha }_{n}}{{M}^{n}}\right)\right\rangle =\ln \sum \limits _{i}^{}{}{{P}_{i}}\exp \left({{\alpha }_{n}}{{M}_{i}}^{n}\right)=\ln \sum \limits _{i}^{}{}{{e}^{\Psi -\left({{\lambda }_{n}}-{{\alpha }_{n}}\right){{M}_{i}}^{n}}}\\&=\ln \left[{{e}^{\Psi }}\cdot \sum \limits _{i}^{}{}{{e}^{-\left({{\lambda }_{n}}-{{\alpha }_{n}}\right){{M}_{i}}^{n}}}\right]=\Psi \left(\lambda \right)+\ln \left[\sum \limits _{i}^{}{}{{e}^{-\left({{\lambda }_{n}}-{{\alpha }_{n}}\right){{M}_{i}}^{n}}}\right]\\&\ln \left[\sum \limits _{i}^{}{}{{e}^{-\left({{\lambda }_{n}}-{{\alpha }_{n}}\right){{M}_{i}}^{n}}}\right]=-\Psi \left(\lambda -\alpha \right)\\&\Rightarrow \Gamma \left(\alpha \right)=\Psi \left(\lambda \right)-\Psi \left(\lambda -\alpha \right)\\&\Rightarrow {{Q}^{mn}}=-{{\left.{\frac {{{\partial }^{2}}\Psi \left(\lambda -\alpha \right)}{\partial {{\alpha }_{m}}\partial {{\alpha }_{n}}}}\right|}_{\alpha =0}}=-{\frac {{{\partial }^{2}}\Psi \left(\lambda \right)}{\partial {{\lambda }_{m}}\partial {{\lambda }_{n}}}}=-{{\eta }^{mn}}\\\end{aligned}}

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\begin{aligned} Γ (α) = \ln ⟨ \exp (α_{n} M^{n}) ⟩ = \ln \sum_{i}^{} P_{i} \exp (α_{n} {M_{i}}^{n}) = \ln \sum_{i}^{} e^{Ψ - (λ_{n} - α_{n}) {M_{i}}^{n}} \\ = \ln [e^{Ψ} \cdot \sum_{i}^{} e^{- (λ_{n} - α_{n}) {M_{i}}^{n}}] = Ψ (λ) + \ln [\sum_{i}^{} e^{- (λ_{n} - α_{n}) {M_{i}}^{n}}] \\ \ln [\sum_{i}^{} e^{- (λ_{n} - α_{n}) {M_{i}}^{n}}] = - Ψ (λ - α) \\ \Rightarrow Γ (α) = Ψ (λ) - Ψ (λ - α) \\ \Rightarrow Q^{m n} = - {\frac{\partial^{2} Ψ (λ - α)}{\partial α_{m} \partial α_{n}} |}_{α = 0} = - \frac{\partial^{2} Ψ (λ)}{\partial λ_{m} \partial λ_{n}} = - η^{m n} \end{aligned}

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Identifiers

$Γ$

$α$

$α_{n}$

$M$

$n$

$i$

$P_{i}$

$α_{n}$

$M_{i}$

$n$

$i$

$e$

$Ψ$

$λ_{n}$

$α_{n}$

$M_{i}$

$n$

$e$

$Ψ$

$i$

$e$

$λ_{n}$

$α_{n}$

$M_{i}$

$n$

$Ψ$

$λ$

$i$

$e$

$λ_{n}$

$α_{n}$

$M_{i}$

$n$

$i$

$e$

$λ_{n}$

$α_{n}$

$M_{i}$

$n$

$Ψ$

$λ$

$α$

$Γ$

$α$

$Ψ$

$λ$

$Ψ$

$λ$

$α$

$Q$

$m$

$n$

$Ψ$

$λ$

$α$

$α_{m}$

$α_{n}$

$α$

$Ψ$

$λ$

$λ_{m}$

$λ_{n}$

$η$

$m$

$n$

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