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Display information for equation id:math.939.26 on revision:939

* Page found: Verallgemeinerte kanonische Verteilung (eq math.939.26)

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Verallgemeinerte kanonische Verteilung Eq: math.940.26

Hash: 8bf6bc244f0d3f6f9da40d45f1d7a3b8

TeX (original user input):

\begin{align}
  & \delta I(\rho )=\int_{{}}^{{}}{{{d}^{d}}x\left( \ln \rho (x)+1 \right)\delta \rho (x)}=0 \\ 
 & \Rightarrow \int_{{}}^{{}}{{{d}^{d}}x\delta \rho (x)}=0 \\ 
 & \int_{{}}^{{}}{{{d}^{d}}x{{M}^{n}}(x)\delta \rho (x)}=0 \\ 
 & \Rightarrow \int_{{}}^{{}}{{{d}^{d}}x\left( \ln \rho -\Psi +{{\lambda }_{n}}{{M}^{n}} \right)\delta \rho (x)}=0 \\ 
 & \Rightarrow \rho (x)=\exp (\Psi -{{\lambda }_{n}}{{M}^{n}}) \\ 
\end{align}

TeX (checked):

{\begin{aligned}&\delta I(\rho )=\int _{}^{}{{{d}^{d}}x\left(\ln \rho (x)+1\right)\delta \rho (x)}=0\\&\Rightarrow \int _{}^{}{{{d}^{d}}x\delta \rho (x)}=0\\&\int _{}^{}{{{d}^{d}}x{{M}^{n}}(x)\delta \rho (x)}=0\\&\Rightarrow \int _{}^{}{{{d}^{d}}x\left(\ln \rho -\Psi +{{\lambda }_{n}}{{M}^{n}}\right)\delta \rho (x)}=0\\&\Rightarrow \rho (x)=\exp(\Psi -{{\lambda }_{n}}{{M}^{n}})\\\end{aligned}}

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\begin{matrix} δ I (ρ) = \int_{}^{} d^{d} x (\ln ρ (x) + 1) δ ρ (x) = 0 \\ \Rightarrow \int_{}^{} d^{d} x δ ρ (x) = 0 \\ \int_{}^{} d^{d} x M^{n} (x) δ ρ (x) = 0 \\ \Rightarrow \int_{}^{} d^{d} x (\ln ρ - Ψ + λ_{n} M^{n}) δ ρ (x) = 0 \\ \Rightarrow ρ (x) = \exp (Ψ - λ_{n} M^{n}) \end{matrix}

<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>δ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ln</mi><mo>&#x2061;</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>δ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mi>δ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>δ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ln</mi><mo>&#x2061;</mo><mi>ρ</mi><mo stretchy="false">−</mo><mi>Ψ</mi><mo stretchy="false">+</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>δ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>exp</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>Ψ</mi><mo stretchy="false">−</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$δ$

$I$

$ρ$

$d$

$x$

$ρ$

$x$

$δ$

$ρ$

$x$

$d$

$x$

$δ$

$ρ$

$x$

$d$

$x$

$M$

$n$

$x$

$δ$

$ρ$

$x$

$d$

$x$

$ρ$

$Ψ$

$λ_{n}$

$M$

$n$

$δ$

$ρ$

$x$

$ρ$

$x$

$Ψ$

$λ_{n}$

$M$

$n$

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