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<title>Zwanzig-Projektionsoperator</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Zwanzig-Projektionsoperator</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Der Zwanzig-Projektionsoperator ist ein mathematisches Werkzeug aus der <a href="Statistische_Mechanik" title="Statistische Mechanik">Statistischen Mechanik</a>.<a href="#cite_note-Zwanzig1961-1">[1]</a>

Der <a href="Projektionsoperator" class="mw-redirect" title="Projektionsoperator">Projektionsoperator</a> wirkt im linearen Raum der <a href="Phasenraum" title="Phasenraum">Phasenraum</a>-Funktionen, und projiziert auf den linearen Unterraum der „langsamen“ Phasenraum-Funktionen. Der Operator wurde von <a href="Robert_Zwanzig" title="Robert Zwanzig">Robert Zwanzig</a> eingeführt, um eine generische <a href="Mastergleichung" title="Mastergleichung">Mastergleichung</a> herzuleiten.

Er wird meistens in diesem oder ähnlichem Kontext verwendet, um auf formale Weise Bewegungsgleichungen für gewisse „langsame“ kollektive Variablen herzuleiten.<a href="#cite_note-Kawasaki1973-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Langsame_Variable_und_Skalarprodukt">Langsame Variable und Skalarprodukt</h2></div>

Der Zwanzig-Projektionsoperator wirkt auf Funktionen im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6N}">

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<annotation encoding="application/x-tex">{\displaystyle 6N}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37b183457644de3b21f0eb69d056ce6f33d3580c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 6N}" loading="lazy">-dimensionalen Phasenraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =\{\mathbf {q} _{i},\mathbf {p} _{i}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma =\{\mathbf {q} _{i},\mathbf {p} _{i}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e158bd9996f2687e5af2d9f1e7e49ec9b1516384.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.406ex; height:2.843ex;" alt="{\displaystyle \Gamma =\{\mathbf {q} _{i},\mathbf {p} _{i}\}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> Punktteilchen mit Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} _{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8ce6be9c9335b20681f1b784557c574a69f28e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.211ex; height:2.176ex;" alt="{\displaystyle \mathbf {q} _{i}}" loading="lazy"> und Impulsen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f92664042a291accf70dd087f3dea939ad833419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.285ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{i}}" loading="lazy">.

Eine spezielle Teilmenge dieser Funktionen ist eine aufzählbare Menge von „langsamen Variablen“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\Gamma )=\{A_{n}(\Gamma )\}}">

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<annotation encoding="application/x-tex">{\displaystyle A(\Gamma )=\{A_{n}(\Gamma )\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b332373e263f4b59cabc935542b1347fd1c38394.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.652ex; height:2.843ex;" alt="{\displaystyle A(\Gamma )=\{A_{n}(\Gamma )\}}" loading="lazy">.

Kandidaten für einige dieser Variablen könnten sein die langwelligen Fourierkomponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{k}(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{k}(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ec08d6342b92d2673bb4570dce172a0bab7c7c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.553ex; height:2.843ex;" alt="{\displaystyle \rho _{k}(\Gamma )}" loading="lazy"> der Massendichte und die langwelligen Fourierkomponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\pi } _{\mathbf {k} }(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\pi } _{\mathbf {k} }(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/765dbf175c6c27396cc89c406670aa82012f7b76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.817ex; height:2.843ex;" alt="{\displaystyle \mathbf {\pi } _{\mathbf {k} }(\Gamma )}" loading="lazy"> der Impulsdichte, mit Wellenvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea699cbc1f843f2e855577d57529430ec33a1ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }" loading="lazy"> identifiziert mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

Der Zwanzig-Projektionsoperator verwendet diese Funktionen, liefert aber keine Information darüber, wie man die langsamen Variablen einer <a href="Hamiltonfunktion" class="mw-redirect" title="Hamiltonfunktion">Hamiltonfunktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle H(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cabd830ad1b7adcad7ca4621007d8d4ea2585908.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.326ex; height:2.843ex;" alt="{\displaystyle H(\Gamma )}" loading="lazy"> finden kann.

Ein Skalarprodukt<a href="#cite_note-Mori1965-3">[3]</a>

zwischen zwei beliebigen Phasenraumfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle f_{1}(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7f46a6d420d649b337712ba0a90894c9cd6c155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.456ex; height:2.843ex;" alt="{\displaystyle f_{1}(\Gamma )}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle f_{2}(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f73bebda805656f6df227f99454417a8e65a7517.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.456ex; height:2.843ex;" alt="{\displaystyle f_{2}(\Gamma )}" loading="lazy"> ist definiert durch die Gleichgewichtskorrelation

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<annotation encoding="application/x-tex">{\displaystyle \left(f_{1},f_{2}\right)=\int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)f_{1}\left(\Gamma \right)f_{2}\left(\Gamma \right),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1277c8d1895f59e2c017a5010b61a2cfd0171969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.053ex; height:5.676ex;" alt="{\displaystyle \left(f_{1},f_{2}\right)=\int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)f_{1}\left(\Gamma \right)f_{2}\left(\Gamma \right),}" loading="lazy"></dd></dl>

wobei

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<annotation encoding="application/x-tex">{\displaystyle \rho _{0}\left(\Gamma \right)={\frac {\delta \left(H\left(\Gamma \right)-E\right)}{\int \mathrm {d} \Gamma ^{\prime }\delta \left(H\left(\Gamma ^{\prime }\right)-E\right)}},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1a172d14c3f5e87b6484de55577ff44395d1933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.98ex; height:6.676ex;" alt="{\displaystyle \rho _{0}\left(\Gamma \right)={\frac {\delta \left(H\left(\Gamma \right)-E\right)}{\int \mathrm {d} \Gamma ^{\prime }\delta \left(H\left(\Gamma ^{\prime }\right)-E\right)}},}" loading="lazy"></dd></dl>

die <a href="Mikrokanonisches_Ensemble" title="Mikrokanonisches Ensemble">mikrokanonische</a> Gleichgewichtsverteilung bezeichnet. „Schnelle“ Variablen sind per Definition unter diesem Skalarprodukt orthogonal zu allen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(A(\Gamma ))}">

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<annotation encoding="application/x-tex">{\displaystyle G(A(\Gamma ))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4d8d01557968ed8759a87f748513c455304d989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.641ex; height:2.843ex;" alt="{\displaystyle G(A(\Gamma ))}" loading="lazy"> der „langsamen“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle A(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7ee9fe601b2b2c6953c630015d9f329719c238e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:2.843ex;" alt="{\displaystyle A(\Gamma )}" loading="lazy">.

Diese Definition besagt, dass Fluktuationen schneller und langsamer Variablen unkorreliert sind, und gemäß <a href="Ergodenhypothese" title="Ergodenhypothese">Ergodenhypothese</a> gilt dies auch für das Zeitmittel.

Wenn eine generische Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12746c21fe65fae91efb1f4f468b73521cb5d32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle f(\Gamma )}" loading="lazy"> mit langsamen Variablen korreliert ist, dann kann man davon Funktionen langsamer Variablen subtrahieren, bis nur mehr der unkorrelierte schnelle Anteil von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12746c21fe65fae91efb1f4f468b73521cb5d32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle f(\Gamma )}" loading="lazy"> verbleibt.

Das Produkt einer langsamen und einer schnellen Variable ist eine schnelle Variable.

<div class="mw-heading mw-heading2"><h2 id="Der_Projektionsoperator">Der Projektionsoperator</h2></div>

Betrachte das Kontinuum von Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{a}(\Gamma )=\delta (A(\Gamma )-a)=\prod _{n}\delta (A_{n}(\Gamma )-a_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{a}(\Gamma )=\delta (A(\Gamma )-a)=\prod _{n}\delta (A_{n}(\Gamma )-a_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6f706801362bb884f06d28746f44cc989f549dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.899ex; height:5.509ex;" alt="{\displaystyle \Phi _{a}(\Gamma )=\delta (A(\Gamma )-a)=\prod _{n}\delta (A_{n}(\Gamma )-a_{n})}" loading="lazy"> mit konstantem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=a_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a=a_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ae1f1b5bd691dca73f3c0e5b06627a65b5240cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle a=a_{n}}" loading="lazy">.

Jede Phasenraumfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(A(\Gamma ))}">

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<annotation encoding="application/x-tex">{\displaystyle G(A(\Gamma ))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4d8d01557968ed8759a87f748513c455304d989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.641ex; height:2.843ex;" alt="{\displaystyle G(A(\Gamma ))}" loading="lazy">, die von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> nur über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle A(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7ee9fe601b2b2c6953c630015d9f329719c238e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:2.843ex;" alt="{\displaystyle A(\Gamma )}" loading="lazy"> abhängt, ist eine Funktion der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{a}}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{a}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d65ceee07e21d4444f84d8a5abfe84bff07926f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.78ex; height:2.509ex;" alt="{\displaystyle \Phi _{a}}" loading="lazy">, nämlich

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<annotation encoding="application/x-tex">{\displaystyle G(A\left(\Gamma \right))=\int \mathrm {d} aG\left(a\right)\delta \left(A\left(\Gamma \right)-a\right).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0ff64f899f4e91f7b41c497386d7a0426b86e37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.611ex; height:5.676ex;" alt="{\displaystyle G(A\left(\Gamma \right))=\int \mathrm {d} aG\left(a\right)\delta \left(A\left(\Gamma \right)-a\right).}" loading="lazy"></dd></dl>

Eine generische Phasenraumfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12746c21fe65fae91efb1f4f468b73521cb5d32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle f(\Gamma )}" loading="lazy"> lässt sich daher schreiben

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<annotation encoding="application/x-tex">{\displaystyle f\left(\Gamma \right)=F\left(A\left(\Gamma \right)\right)+R\left(\Gamma \right),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b614df588f62a6fd09f32c7de1f8593bc69908bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.643ex; height:2.843ex;" alt="{\displaystyle f\left(\Gamma \right)=F\left(A\left(\Gamma \right)\right)+R\left(\Gamma \right),}" loading="lazy"></dd></dl>

wo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle R(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c90e3910e29e8d897c02d2fe59e7b778b21ca54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.026ex; height:2.843ex;" alt="{\displaystyle R(\Gamma )}" loading="lazy"> der schnelle Teil von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12746c21fe65fae91efb1f4f468b73521cb5d32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle f(\Gamma )}" loading="lazy"> ist.

Einen Ausdruck für den langsamen Teil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\Gamma )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle F(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0ab81872723ecd37fa62591cd31a289c94b9eb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.003ex; height:2.843ex;" alt="{\displaystyle F(\Gamma )}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> erhält man, wenn man das Skalarprodukt mit der langsamen Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (A(\Gamma )-a)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \delta (A(\Gamma )-a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12f412a653a745ff9d75ab42da2124b5a8209c6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.933ex; height:2.843ex;" alt="{\displaystyle \delta (A(\Gamma )-a)}" loading="lazy"> bildet,

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<mrow>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)f\left(\Gamma \right)\delta \left(A\left(\Gamma \right)-a\right)=\int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)F\left(A\left(\Gamma \right)\right)\delta \left(A\left(\Gamma \right)-a\right)=F\left(a\right)\int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)\delta \left(A\left(\Gamma \right)-a\right).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19a009f99f9f1ea439af8e0c10e75c939987f389.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:100.795ex; height:5.676ex;" alt="{\displaystyle \int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)f\left(\Gamma \right)\delta \left(A\left(\Gamma \right)-a\right)=\int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)F\left(A\left(\Gamma \right)\right)\delta \left(A\left(\Gamma \right)-a\right)=F\left(a\right)\int \mathrm {d} \Gamma \rho _{0}\left(\Gamma \right)\delta \left(A\left(\Gamma \right)-a\right).}" loading="lazy"></dd></dl>

Dies liefert einen Ausdruck für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\Gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(\Gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0ab81872723ecd37fa62591cd31a289c94b9eb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.003ex; height:2.843ex;" alt="{\displaystyle F(\Gamma )}" loading="lazy">, und somit für den Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">,

welcher eine beliebige Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\Gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12746c21fe65fae91efb1f4f468b73521cb5d32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle f(\Gamma )}" loading="lazy"> auf ihren langsamen Teil projiziert, abhängig von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> nur über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\Gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(\Gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7ee9fe601b2b2c6953c630015d9f329719c238e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:2.843ex;" alt="{\displaystyle A(\Gamma )}" loading="lazy">,

<mrow>

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<mrow>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle P\cdot f\left(\Gamma \right)=F\left(A\left(\Gamma \right)\right)={\frac {\int \mathrm {d} \Gamma ^{\prime }\rho _{0}\left(\Gamma ^{\prime }\right)f\left(\Gamma ^{\prime }\right)\delta \left(A\left(\Gamma ^{\prime }\right)-A\left(\Gamma \right)\right)}{\int \mathrm {d} \Gamma ^{\prime }\rho _{0}\left(\Gamma ^{\prime }\right)\delta \left(A\left(\Gamma ^{\prime }\right)-A\left(\Gamma \right)\right)}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80cbf15e9f124053d5e1a2d66e34ac6dfd9bb50d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:61.128ex; height:6.843ex;" alt="{\displaystyle P\cdot f\left(\Gamma \right)=F\left(A\left(\Gamma \right)\right)={\frac {\int \mathrm {d} \Gamma ^{\prime }\rho _{0}\left(\Gamma ^{\prime }\right)f\left(\Gamma ^{\prime }\right)\delta \left(A\left(\Gamma ^{\prime }\right)-A\left(\Gamma \right)\right)}{\int \mathrm {d} \Gamma ^{\prime }\rho _{0}\left(\Gamma ^{\prime }\right)\delta \left(A\left(\Gamma ^{\prime }\right)-A\left(\Gamma \right)\right)}}.}" loading="lazy"></dd></dl>

Dieser Ausdruck stimmt mit dem Ausdruck von Zwanzig,<a href="#cite_note-Zwanzig1961-1">[1]</a> überein, außer dass Zwanzig die Hamiltonfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\Gamma )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle H(\Gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cabd830ad1b7adcad7ca4621007d8d4ea2585908.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.326ex; height:2.843ex;" alt="{\displaystyle H(\Gamma )}" loading="lazy"> mit zu den langsamen Variablen zählt. Der Zwanzig-Projektionsoperator erfüllt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PG(A(\Gamma ))=G(A(\Gamma ))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PG(A(\Gamma ))=G(A(\Gamma ))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b830e13ec2a23dbd70ea487878f714faa96e97aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.126ex; height:2.843ex;" alt="{\displaystyle PG(A(\Gamma ))=G(A(\Gamma ))}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{2}=P}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle P^{2}=P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b0aaa12409b3907e6b00649b654249d5d4e82ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.72ex; height:2.676ex;" alt="{\displaystyle P^{2}=P}" loading="lazy">. Der schnelle Teil von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\Gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12746c21fe65fae91efb1f4f468b73521cb5d32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle f(\Gamma )}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-P)f(\Gamma )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (1-P)f(\Gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f8ba244d7c5936b20b15e0c4e877b6e95849ecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.098ex; height:2.843ex;" alt="{\displaystyle (1-P)f(\Gamma )}" loading="lazy">. Funktionen langsamer Variablen und insbesondere Produkte von langsamen Variablen sind langsame Variablen. Der Raum der langsamen Variablen ist somit eine Algebra. Die Algebra ist i. A. nicht abgeschlossen unter der <a href="Poissonklammer" class="mw-redirect" title="Poissonklammer">Poissonklammer</a>, inklusive der Poissonklammer mit der Hamiltonfunktion.

<div class="mw-heading mw-heading2"><h2 id="Bezug_zu_Liouvillegleichung_und_Mastergleichungen">Bezug zu Liouvillegleichung und Mastergleichungen</h2></div>

Die Motivation für die Definition des Skalarprodukts und des Projektionsoperators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> ist letztendlich, dass es damit möglich ist, eine <a href="Mastergleichung" title="Mastergleichung">Mastergleichung</a> für die zeitabhängige Wahrscheinlichkeitsverteilung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(a,t)}">

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<annotation encoding="application/x-tex">{\displaystyle p(a,t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/466f5a994c8cd969ccc77587587b4fe3790aa141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.172ex; height:2.843ex;" alt="{\displaystyle p(a,t)}" loading="lazy"> der langsamen Variablen (oder eine <a href="Langevingleichung" class="mw-redirect" title="Langevingleichung">Langevingleichungen</a> für die langsamen Variablen selber) herzuleiten.

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\Gamma ,t)=\rho _{0}(\Gamma )\sigma (\Gamma ,t)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \rho (\Gamma ,t)=\rho _{0}(\Gamma )\sigma (\Gamma ,t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60dc7899e59872debbd521ea7bac1c158882c72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.42ex; height:2.843ex;" alt="{\displaystyle \rho (\Gamma ,t)=\rho _{0}(\Gamma )\sigma (\Gamma ,t)}" loading="lazy">

die zeitabhängige Wahrscheinlichkeitsverteilung im Phasenraum. Die Phasenraumfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (\Gamma ,t)}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (\Gamma ,t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d64aa121e31556dcc83961cf712aac275f09ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.465ex; height:2.843ex;" alt="{\displaystyle \sigma (\Gamma ,t)}" loading="lazy"> ist (ebenso wie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\Gamma ,t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (\Gamma ,t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8acc91f6bdd74e38f38ed39d715c0cf0442cf9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.338ex; height:2.843ex;" alt="{\displaystyle \rho (\Gamma ,t)}" loading="lazy">) eine Lösung der <a href="Liouville-Gleichung" title="Liouville-Gleichung">Liouvillegleichung</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i{\frac {\partial }{\partial t}}\sigma (\Gamma ,t)=L\sigma (\Gamma ,t).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/560ad9f8334b4e086799cfb4112d1d56e7216de8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.055ex; height:5.509ex;" alt="{\displaystyle i{\frac {\partial }{\partial t}}\sigma (\Gamma ,t)=L\sigma (\Gamma ,t).}" loading="lazy"></dd></dl>

Der wesentliche Schritt ist dann zu schreiben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}=P\sigma }">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \rho _{1}=P\sigma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3801a54169afa46604921886e607107b21612fbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.43ex; height:2.676ex;" alt="{\displaystyle \rho _{1}=P\sigma }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{2}=(1-P)\sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{2}=(1-P)\sigma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6f6a749466b7e24803cc616a135c3d5e2f8fb77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.242ex; height:2.843ex;" alt="{\displaystyle \rho _{2}=(1-P)\sigma }" loading="lazy">, und die Liouvillegleichung auf den schnellen und langsamen Unterraum zu projizieren,<a href="#cite_note-Zwanzig1961-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle i{\frac {\partial }{\partial t}}\rho _{1}=PL\rho _{1}+PL\rho _{2},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98a8ac70e490fb435e5b0f159980edc9fb4b4073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.807ex; height:5.509ex;" alt="{\displaystyle i{\frac {\partial }{\partial t}}\rho _{1}=PL\rho _{1}+PL\rho _{2},}" loading="lazy"></dd>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle i{\frac {\partial }{\partial t}}\rho _{2}=\left(1-P\right)L\rho _{2}+\left(1-P\right)L\rho _{1}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bf0db94482e68c970041e7ca775afe945d6fe4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.206ex; height:5.509ex;" alt="{\displaystyle i{\frac {\partial }{\partial t}}\rho _{2}=\left(1-P\right)L\rho _{2}+\left(1-P\right)L\rho _{1}.}" loading="lazy"></dd></dl>

Wenn man dann die zweite Gleichung nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/793b211571b3ffe34c4639654d567296d29d7f72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{2}}" loading="lazy"> auflöst und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{2}(\Gamma ,t)}">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{2}(\Gamma ,t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81def87fde05273e23fa2796ae415f3209899fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.392ex; height:2.843ex;" alt="{\displaystyle \rho _{2}(\Gamma ,t)}" loading="lazy"> in die erste Gleichung einsetzt, ergibt sich eine Gleichung für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e0f2d347f2a0ed7f7c9808c427a89813b957017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{1}}" loading="lazy"> (siehe <a href="Nakajima-Zwanzig-Gleichung" title="Nakajima-Zwanzig-Gleichung">Nakajima-Zwanzig-Gleichung</a>). Die letzte Gleichung schließlich ergibt eine Gleichung für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(A(\Gamma ),t)=p_{0}(A(\Gamma ))\rho _{1}(\Gamma ,t),}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p(A(\Gamma ),t)=p_{0}(A(\Gamma ))\rho _{1}(\Gamma ,t),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d631d3a2135f5f1b89b88f68a823e1f705253af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:30.122ex; height:2.843ex;" alt="{\displaystyle p(A(\Gamma ),t)=p_{0}(A(\Gamma ))\rho _{1}(\Gamma ,t),}" loading="lazy"> wo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}(a)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{0}(a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98c9f2537e6ef3e5f0b0e3c3106bd61f86ab6497.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.352ex; height:2.843ex;" alt="{\displaystyle p_{0}(a)}" loading="lazy"> die Gleichgewichtsverteilung der langsamen Variablen bezeichnet.

<div class="mw-heading mw-heading2"><h2 id="Nichtlineare_Langevingleichungen">Nichtlineare Langevingleichungen</h2></div>

Der Ausgangspunkt für die Standard-Herleitung einer Langevingleichung ist die Identität <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=P+Q}">

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<annotation encoding="application/x-tex">{\displaystyle 1=P+Q}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71200327ff299c696b65710f0c6d600bb5b8baac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.685ex; height:2.509ex;" alt="{\displaystyle 1=P+Q}" loading="lazy">, wo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> in den schnellen Unterraum projiziert. Betrachte diskrete kleine Zeitschritte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> mit Evolutionsoperator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cong 1+i\tau L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\cong 1+i\tau L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0be8dc4b87fa4e1c50a4fcfaf886507e79185be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.471ex; height:2.343ex;" alt="{\displaystyle U\cong 1+i\tau L}" loading="lazy">, wo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> der <a href="Liouville-Gleichung" title="Liouville-Gleichung">Liouville-Operator</a> ist.

Das Ziel ist es, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle U^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8daca69ac38435404c76e140c9b94a2f873c95aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.06ex; height:2.343ex;" alt="{\displaystyle U^{n}}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{k}P}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle U^{k}P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49dd9217a0d410cfb26aaeafc438cf335173ee0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.675ex; height:2.676ex;" alt="{\displaystyle U^{k}P}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(UQ)^{m}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(UQ)^{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48701c3b60bcb0f310542714f2ef70a6e3bde6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.944ex; height:2.843ex;" alt="{\displaystyle Q(UQ)^{m}}" loading="lazy"> auszudrücken.

Die Motivation dafür ist, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{k}P}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U^{k}P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49dd9217a0d410cfb26aaeafc438cf335173ee0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.675ex; height:2.676ex;" alt="{\displaystyle U^{k}P}" loading="lazy"> ein Funktional von langsamen Variablen ist, während <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(UQ)^{m}}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle Q(UQ)^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48701c3b60bcb0f310542714f2ef70a6e3bde6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.944ex; height:2.843ex;" alt="{\displaystyle Q(UQ)^{m}}" loading="lazy"> Ausdrücke erzeugt, welche zu jedem Zeitpunkt schnelle Variablen sind. Die Erwartung ist, diese schnellen Variablen durch irgendwelche Modelldaten repräsentierbar sind, z. B. durch ein Gaußsches weißes Rauschen. Die Zerlegung erreicht man, indem man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=P+Q}">

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<annotation encoding="application/x-tex">{\displaystyle 1=P+Q}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71200327ff299c696b65710f0c6d600bb5b8baac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.685ex; height:2.509ex;" alt="{\displaystyle 1=P+Q}" loading="lazy"> von links mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> multipliziert, außer für den letzten Term,

welcher mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=PU+QU}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=PU+QU}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a7bfaf9147c05afe8dde7f1f40469ec50652c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.87ex; height:2.509ex;" alt="{\displaystyle U=PU+QU}" loading="lazy"> multipliziert wird. Iteration ergibt

<mtr>

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</mrow>

</munderover>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1&=P+Q,\\U&=UP+PUQ+QUQ,\\...&=...\\U^{n}&=U^{n}P+\sum _{m=1}^{n}U^{n-m}P\left(UQ\right)^{m}+Q\left(UQ\right)^{n}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6c568bffeeee8b123a453e033064fb346d768a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:45.092ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}1&=P+Q,\\U&=UP+PUQ+QUQ,\\...&=...\\U^{n}&=U^{n}P+\sum _{m=1}^{n}U^{n-m}P\left(UQ\right)^{m}+Q\left(UQ\right)^{n}.\end{aligned}}}" loading="lazy"></dd></dl>

Die letzte Zeile lässt sich auch per Induktion beweisen. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=1+itL/n}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=1+itL/n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/143cb3e8c55c875accf00c2dade6f1d6493e83cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.666ex; height:2.843ex;" alt="{\displaystyle U=1+itL/n}" loading="lazy"> führt der Limes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }" loading="lazy"> direkt auf die Operator-Identität von Kawasaki<a href="#cite_note-Kawasaki1973-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle e^{itL}=e^{itL}P+i\int _{0}^{t}\mathrm {d} se^{i\left(t-s\right)L}PLQe^{isLQ}+Qe^{itLQ}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e5cf660b1b028804f872345f8dc10980c8c2501.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.513ex; height:6.176ex;" alt="{\displaystyle e^{itL}=e^{itL}P+i\int _{0}^{t}\mathrm {d} se^{i\left(t-s\right)L}PLQe^{isLQ}+Qe^{itLQ}.}" loading="lazy"></dd></dl>

Eine generische Langevingleichungen ergibt sich durch Anwendung dieser Gleichung auf die <a href="Zeitableitung" title="Zeitableitung">Zeitableitung</a> einer langsamen Variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dA(\Gamma ,t)/dt=e^{itL}(dA(\Gamma ,t)/dt)_{t=0}}">

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<annotation encoding="application/x-tex">{\displaystyle dA(\Gamma ,t)/dt=e^{itL}(dA(\Gamma ,t)/dt)_{t=0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76a0835443340dda6ad5f232232202744837a016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.055ex; height:3.176ex;" alt="{\displaystyle dA(\Gamma ,t)/dt=e^{itL}(dA(\Gamma ,t)/dt)_{t=0}}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dA}{dt}}\left(\Gamma ,t\right)&=V+K+R,\\V&=e^{itL}P{\dot {A}}\left(\Gamma ,0\right),\\K&=i\int _{0}^{t}\mathrm {d} se^{i\left(t-s\right)L}PLQe^{isLQ}{\dot {A}}\left(\Gamma ,0\right)=i\int _{0}^{t}\mathrm {d} se^{i\left(t-s\right)L}PLR\left(s\right),\\R&=Qe^{itLQ}{\dot {A}}\left(\Gamma ,0\right).\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be1cd1be6a0dff727aeff1cfceb1ceccab44252d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.55ex; margin-bottom: -0.288ex; width:71.89ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {dA}{dt}}\left(\Gamma ,t\right)&=V+K+R,\\V&=e^{itL}P{\dot {A}}\left(\Gamma ,0\right),\\K&=i\int _{0}^{t}\mathrm {d} se^{i\left(t-s\right)L}PLQe^{isLQ}{\dot {A}}\left(\Gamma ,0\right)=i\int _{0}^{t}\mathrm {d} se^{i\left(t-s\right)L}PLR\left(s\right),\\R&=Qe^{itLQ}{\dot {A}}\left(\Gamma ,0\right).\end{aligned}}}" loading="lazy"></dd></dl>

Hier ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> die (nur von schnellen Variablen abhängende) fluktuierende Kraft. Der Modenkopplungsterme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> und Dämpfungsterme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> sind Funktionale von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c0905fbce73d8bde912fb882c351e336ab56cca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.392ex; height:2.843ex;" alt="{\displaystyle A(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(t-s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(t-s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa0bb0e20e9239e480602f01f172ce4827e05b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.323ex; height:2.843ex;" alt="{\displaystyle A(t-s)}" loading="lazy"> und lassen sich vereinfachen.<a href="#cite_note-Zwanzig1961-1">[1]</a><a href="#cite_note-Kawasaki1973-2">[2]</a><a href="#cite_note-Gunton1979-4">[4]</a><a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Diskrete_Funktionsmenge,_Bezug_zum_Mori-Projektionsoperator">Diskrete Funktionsmenge, Bezug zum Mori-Projektionsoperator</h2></div>

Anstatt den langsamen Teil von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12746c21fe65fae91efb1f4f468b73521cb5d32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle f(\Gamma )}" loading="lazy"> nach dem Kontinuum von Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{a}(\Gamma )=\delta (A(\Gamma )-a)}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{a}(\Gamma )=\delta (A(\Gamma )-a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a24f1abf48bd4756cf5f164ff6ce05403ef2041d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.074ex; height:2.843ex;" alt="{\displaystyle \Phi _{a}(\Gamma )=\delta (A(\Gamma )-a)}" loading="lazy"> zu entwickeln, könnte man auch eine aufzählbare Menge von Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(A(\Gamma ))}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(A(\Gamma ))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57a62f8a14ed24fe4333944d1342f261963b0233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.711ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(A(\Gamma ))}" loading="lazy"> verwenden.

Wenn diese Funktionen ein vollständiges Orthonormalsystem bilden, dann hat der Projektionsoperator die einfache Form

<mrow>

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</munder>

<mrow>

<mrow>

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</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mrow>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P\cdot f\left(\Gamma \right)=\sum _{n}\left(f,\Phi _{n}\right)\Phi _{n}\left(A\left(\Gamma \right)\right).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3413214da7b4158e005a34a61c71e6ecfd50e943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.117ex; height:5.509ex;" alt="{\displaystyle P\cdot f\left(\Gamma \right)=\sum _{n}\left(f,\Phi _{n}\right)\Phi _{n}\left(A\left(\Gamma \right)\right).}" loading="lazy"></dd></dl>

Eine spezielle Wahl für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(A(\Gamma ))}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(A(\Gamma ))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57a62f8a14ed24fe4333944d1342f261963b0233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.711ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(A(\Gamma ))}" loading="lazy"> sind orthonormalisierte Linearkombinationen der langsamen Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\Gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle A(\Gamma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7ee9fe601b2b2c6953c630015d9f329719c238e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:2.843ex;" alt="{\displaystyle A(\Gamma )}" loading="lazy">.

Dies ergibt den <a href="Mori-Zwanzig-Formalismus" title="Mori-Zwanzig-Formalismus">Mori-Projektionsoperator</a>.<a href="#cite_note-Mori1965-3">[3]</a>

Der Satz der linearen Funktionen ist jedoch nicht vollständig, und die orthogonalen Variablen sind nicht schnell oder zufällig, wenn Nichtlinearität in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ins Spiel kommt.

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-Zwanzig1961-1">↑ <a href="#cite_ref-Zwanzig1961_1-0">a</a> <a href="#cite_ref-Zwanzig1961_1-1">b</a> <a href="#cite_ref-Zwanzig1961_1-2">c</a> <a href="#cite_ref-Zwanzig1961_1-3">d</a> Robert Zwanzig: <cite style="font-style:italic">Memory Effects in Irreversible Thermodynamics</cite>. In: <cite style="font-style:italic">Phys. Rev.</cite> 124. Jahrgang, Nr. 4, 1961, S. 983–992, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103/physrev.124.983">10.1103/physrev.124.983</a>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1961PhRv..124..983Z">1961PhRv..124..983Z</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Zwanzig-Projektionsoperator&rft.atitle=Memory+Effects+in+Irreversible+Thermodynamics&rft.au=Robert%26%2332%3BZwanzig&rft.date=1961&rft.doi=10.1103%2Fphysrev.124.983&rft.genre=journal&rft.issue=4&rft.jtitle=Phys.+Rev.&rft.pages=983-992&rft.volume=124.+Jahrgang" style="display:none">

</li>

<li id="cite_note-Kawasaki1973-2">↑ <a href="#cite_ref-Kawasaki1973_2-0">a</a> <a href="#cite_ref-Kawasaki1973_2-1">b</a> <a href="#cite_ref-Kawasaki1973_2-2">c</a> K. Kawasaki: <cite style="font-style:italic">Simple derivations of generalized linear and nonlinear Langevin equations</cite>. In: <cite style="font-style:italic">J. Phys. A: Math. Nucl. Gen.</cite> 6. Jahrgang, Nr. 9, 1973, S. 1289–1295, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088/0305-4470%2F6%2F9%2F004">10.1088/0305-4470/6/9/004</a>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973JPhA....6.1289K">1973JPhA....6.1289K</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Zwanzig-Projektionsoperator&rft.atitle=Simple+derivations+of+generalized+linear+and+nonlinear+Langevin+equations&rft.au=K.%26%2332%3BKawasaki&rft.date=1973&rft.doi=10.1088%2F0305-4470%2F6%2F9%2F004&rft.genre=journal&rft.issue=9&rft.jtitle=J.+Phys.+A%3A+Math.+Nucl.+Gen.&rft.pages=1289-1295&rft.volume=6.+Jahrgang" style="display:none">

</li>

<li id="cite_note-Mori1965-3">↑ <a href="#cite_ref-Mori1965_3-0">a</a> <a href="#cite_ref-Mori1965_3-1">b</a> H. Mori: <cite style="font-style:italic">Transport, Collective Motion, and Brownian Motion</cite>. In: <cite style="font-style:italic">Prog. Theor. Phys.</cite> 33. Jahrgang, Nr. 3, 1965, S. 423–455, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143/ptp.33.423">10.1143/ptp.33.423</a>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1965PThPh..33..423M">1965PThPh..33..423M</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Zwanzig-Projektionsoperator&rft.atitle=Transport%2C+Collective+Motion%2C+and+Brownian+Motion&rft.au=H.%26%2332%3BMori&rft.date=1965&rft.doi=10.1143%2Fptp.33.423&rft.genre=journal&rft.issue=3&rft.jtitle=Prog.+Theor.+Phys.&rft.pages=423-455&rft.volume=33.+Jahrgang" style="display:none">

</li>

<li id="cite_note-Gunton1979-4"><a href="#cite_ref-Gunton1979_4-0">↑</a> J.D. Gunton: <cite style="font-style:italic">Mode coupling theory in relation to the dynamical renormalization group method</cite>. In: <cite style="font-style:italic">Lecture Notes in Physics</cite>. 104. Jahrgang, 1979, S. 1–24, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/3-540-09523-3_1">10.1007/3-540-09523-3_1</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Zwanzig-Projektionsoperator&rft.atitle=Mode+coupling+theory+in+relation+to+the+dynamical+renormalization+group+method&rft.au=J.D.%26%2332%3BGunton&rft.btitle=Lecture+Notes+in+Physics&rft.date=1979&rft.doi=10.1007%2F3-540-09523-3_1&rft.genre=book&rft.pages=1-24&rft.volume=104.+Jahrgang" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> R. Dengler: Another derivation of generalized Langevin equations. <a href="ArXiv" title="ArXiv">arxiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1506.02650v2">1506.02650v2</a>

</li>

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