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Display information for equation id:math.2497.40 on revision:2497

* Page found: Thermodynamische Stabilität (eq math.2497.40)

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Thermodynamische Stabilität Eq: math.2497.40

Hash: 737c3463c4dfa9f9b7b3a77e503e41b3

TeX (original user input):

\begin{align}

& \rho ={{e}^{\left( \Psi -{{\lambda }_{\nu }}{{M}^{\nu }} \right)}} \\

& \Rightarrow b=-\Psi +{{\lambda }_{\nu }}{{M}^{\nu }} \\

& \Rightarrow \Delta b:=b-\left\langle b \right\rangle ={{\lambda }_{\nu }}\left( {{M}^{\nu }}-\left\langle {{M}^{\nu }} \right\rangle  \right)={{\lambda }_{\nu }}\Delta {{M}^{\nu }} \\

& \left\langle {{\left( \Delta b \right)}^{2}} \right\rangle ={{\lambda }_{\nu }}{{\lambda }_{\mu }}\left\langle \Delta {{M}^{\nu }}\Delta {{M}^{\mu }} \right\rangle  \\

\end{align}

TeX (checked):

{\begin{aligned}&\rho ={{e}^{\left(\Psi -{{\lambda }_{\nu }}{{M}^{\nu }}\right)}}\\&\Rightarrow b=-\Psi +{{\lambda }_{\nu }}{{M}^{\nu }}\\&\Rightarrow \Delta b:=b-\left\langle b\right\rangle ={{\lambda }_{\nu }}\left({{M}^{\nu }}-\left\langle {{M}^{\nu }}\right\rangle \right)={{\lambda }_{\nu }}\Delta {{M}^{\nu }}\\&\left\langle {{\left(\Delta b\right)}^{2}}\right\rangle ={{\lambda }_{\nu }}{{\lambda }_{\mu }}\left\langle \Delta {{M}^{\nu }}\Delta {{M}^{\mu }}\right\rangle \\\end{aligned}}

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MathML (2.932 KB / 447 B) :

\begin{matrix} ρ = e^{(Ψ - λ_{ν} M^{ν})} \\ \Rightarrow b = - Ψ + λ_{ν} M^{ν} \\ \Rightarrow Δ b := b - ⟨ b ⟩ = λ_{ν} (M^{ν} - ⟨ M^{ν} ⟩) = λ_{ν} Δ M^{ν} \\ ⟨ {(Δ b)}^{2} ⟩ = λ_{ν} λ_{μ} ⟨ Δ M^{ν} Δ M^{μ} ⟩ \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>ρ</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Ψ</mi><mo stretchy="false">−</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mi>b</mi><mo stretchy="false">=</mo><mo stretchy="false">−</mo><mi>Ψ</mi><mo stretchy="false">+</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mi>Δ</mi><mi>b</mi><mo stretchy="false">:=</mo><mi>b</mi><mo stretchy="false">−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo stretchy="false">−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><mi>Δ</mi><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Δ</mi><mi>b</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>Δ</mi><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mi>Δ</mi><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$ρ$

$e$

$Ψ$

$λ_{ν}$

$M$

$ν$

$b$

$Ψ$

$λ_{ν}$

$M$

$ν$

$Δ$

$b$

$b$

$b$

$λ_{ν}$

$M$

$ν$

$M$

$ν$

$λ_{ν}$

$Δ$

$M$

$ν$

$Δ$

$b$

$λ_{ν}$

$λ_{μ}$

$Δ$

$M$

$ν$

$Δ$

$M$

$μ$

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