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Display information for equation id:math.940.69 on revision:940
* Page found: Verallgemeinerte kanonische Verteilung (eq math.940.69)
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TeX (original user input):
\begin{align}
& \delta \Psi =\frac{\partial \Psi }{\partial {{\lambda }_{n}}}\delta {{\lambda }_{n}}+\frac{1}{2}\frac{{{\partial }^{2}}\Psi }{\partial {{\lambda }_{n}}\partial {{\lambda }_{m}}}\delta {{\lambda }_{n}}\delta {{\lambda }_{m}}+.... \\
& \delta \left\langle {{M}^{n}} \right\rangle =\frac{\partial \left\langle {{M}^{n}} \right\rangle }{\partial {{\lambda }_{n}}}\delta {{\lambda }_{n}}+\frac{1}{2}\frac{{{\partial }^{2}}\left\langle {{M}^{n}} \right\rangle }{\partial {{\lambda }_{n}}\partial {{\lambda }_{m}}}\delta {{\lambda }_{n}}\delta {{\lambda }_{m}}+.... \\
& \Rightarrow K\left( P+\delta P,P \right)=\delta \Psi -\delta {{\lambda }_{n}}\left( \left\langle {{M}^{n}} \right\rangle +\delta \left\langle {{M}^{n}} \right\rangle \right)=\left( \frac{\partial \Psi }{\partial {{\lambda }_{n}}}\delta {{\lambda }_{n}}-\left\langle {{M}^{n}} \right\rangle \right)\delta {{\lambda }_{n}}+\left( \frac{1}{2}\frac{\partial }{\partial {{\lambda }_{m}}}\frac{\partial \Psi }{\partial {{\lambda }_{n}}}-\frac{\partial \left\langle {{M}^{n}} \right\rangle }{\partial {{\lambda }_{m}}} \right)\delta {{\lambda }_{n}}\delta {{\lambda }_{m}} \\
& \frac{\partial \Psi }{\partial {{\lambda }_{n}}}=\left\langle {{M}^{n}} \right\rangle \Rightarrow \left( \frac{1}{2}\frac{\partial }{\partial {{\lambda }_{m}}}\frac{\partial \Psi }{\partial {{\lambda }_{n}}}-\frac{\partial \left\langle {{M}^{n}} \right\rangle }{\partial {{\lambda }_{m}}} \right)=-\frac{1}{2}\frac{\partial \left\langle {{M}^{n}} \right\rangle }{\partial {{\lambda }_{m}}} \\
& \left( \frac{\partial \Psi }{\partial {{\lambda }_{n}}}\delta {{\lambda }_{n}}-\left\langle {{M}^{n}} \right\rangle \right)=0 \\
& \Rightarrow K\left( P+\delta P,P \right)=-\frac{1}{2}\frac{\partial \left\langle {{M}^{n}} \right\rangle }{\partial {{\lambda }_{m}}}\delta {{\lambda }_{n}}\delta {{\lambda }_{m}} \\
& K\left( P+\delta P,P \right)\ge 0 \\
\end{align}
TeX (checked):
{\begin{aligned}&\delta \Psi ={\frac {\partial \Psi }{\partial {{\lambda }_{n}}}}\delta {{\lambda }_{n}}+{\frac {1}{2}}{\frac {{{\partial }^{2}}\Psi }{\partial {{\lambda }_{n}}\partial {{\lambda }_{m}}}}\delta {{\lambda }_{n}}\delta {{\lambda }_{m}}+....\\&\delta \left\langle {{M}^{n}}\right\rangle ={\frac {\partial \left\langle {{M}^{n}}\right\rangle }{\partial {{\lambda }_{n}}}}\delta {{\lambda }_{n}}+{\frac {1}{2}}{\frac {{{\partial }^{2}}\left\langle {{M}^{n}}\right\rangle }{\partial {{\lambda }_{n}}\partial {{\lambda }_{m}}}}\delta {{\lambda }_{n}}\delta {{\lambda }_{m}}+....\\&\Rightarrow K\left(P+\delta P,P\right)=\delta \Psi -\delta {{\lambda }_{n}}\left(\left\langle {{M}^{n}}\right\rangle +\delta \left\langle {{M}^{n}}\right\rangle \right)=\left({\frac {\partial \Psi }{\partial {{\lambda }_{n}}}}\delta {{\lambda }_{n}}-\left\langle {{M}^{n}}\right\rangle \right)\delta {{\lambda }_{n}}+\left({\frac {1}{2}}{\frac {\partial }{\partial {{\lambda }_{m}}}}{\frac {\partial \Psi }{\partial {{\lambda }_{n}}}}-{\frac {\partial \left\langle {{M}^{n}}\right\rangle }{\partial {{\lambda }_{m}}}}\right)\delta {{\lambda }_{n}}\delta {{\lambda }_{m}}\\&{\frac {\partial \Psi }{\partial {{\lambda }_{n}}}}=\left\langle {{M}^{n}}\right\rangle \Rightarrow \left({\frac {1}{2}}{\frac {\partial }{\partial {{\lambda }_{m}}}}{\frac {\partial \Psi }{\partial {{\lambda }_{n}}}}-{\frac {\partial \left\langle {{M}^{n}}\right\rangle }{\partial {{\lambda }_{m}}}}\right)=-{\frac {1}{2}}{\frac {\partial \left\langle {{M}^{n}}\right\rangle }{\partial {{\lambda }_{m}}}}\\&\left({\frac {\partial \Psi }{\partial {{\lambda }_{n}}}}\delta {{\lambda }_{n}}-\left\langle {{M}^{n}}\right\rangle \right)=0\\&\Rightarrow K\left(P+\delta P,P\right)=-{\frac {1}{2}}{\frac {\partial \left\langle {{M}^{n}}\right\rangle }{\partial {{\lambda }_{m}}}}\delta {{\lambda }_{n}}\delta {{\lambda }_{m}}\\&K\left(P+\delta P,P\right)\geq 0\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi>δ</mi><mi mathvariant="normal">Ψ</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi mathvariant="normal">Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi mathvariant="normal">Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi>δ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>K</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>P</mi><mo>+</mo><mi>δ</mi><mi>P</mi><mo>,</mo><mi>P</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mi>δ</mi><mi mathvariant="normal">Ψ</mi><mo>−</mo><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>+</mo><mi>δ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi mathvariant="normal">Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi mathvariant="normal">Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi mathvariant="normal">Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>⇒</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi mathvariant="normal">Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi mathvariant="normal">Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>K</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>P</mi><mo>+</mo><mi>δ</mi><mi>P</mi><mo>,</mo><mi>P</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mi>δ</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mi>K</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>P</mi><mo>+</mo><mi>δ</mi><mi>P</mi><mo>,</mo><mi>P</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>≥</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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