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* Page found: Verallgemeinerte kanonische Verteilung (eq math.940.67)

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Verallgemeinerte kanonische Verteilung Eq: math.940.67

Hash: 921cf7e74a3a4a2a981118792e2712cf

TeX (original user input):

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

TeX (checked):

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

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MathML (9.663 KB / 615 B) :

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

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