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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ehrenfest-Modell</span></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"><p>Das <b>Ehrenfest-Modell</b> (auch als <b>Ehrenfest-Kette</b> bekannt) ist ein <a href="Stochastik" title="Stochastik">stochastisches</a> Modell, das den Stoffaustausch zwischen zwei durch eine <a href="Membran_(Trennschicht)" title="Membran (Trennschicht)">Membran</a> getrennte Behältnisse beschreibt. Das Modell wurde zuerst durch den <a href="%C3%96sterreich" title="Österreich">österreichischen</a> <a href="Physik" title="Physik">Physiker</a> <a href="Paul_Ehrenfest" title="Paul Ehrenfest">Paul Ehrenfest</a> (<a href="1880" title="1880">1880</a>–<a href="1933" title="1933">1933</a>) vorgeschlagen und ist einer von vielen Beiträgen der Physik zur Entwicklung der mathematischen Theorie der <a href="Stochastischer_Prozess" title="Stochastischer Prozess">stochastischen Prozesse</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Das_Modell">Das Modell</h2></div>
<p>Bei verschiedenen Substanzen wurde beobachtet, dass die Verteilung der Substanz in einem solchen Experiment im Laufe der Zeit zwar einem <a href="Chemisches_Gleichgewicht" title="Chemisches Gleichgewicht">Gleichgewichtszustand</a> entgegenstrebt, aber dennoch auch nach Erreichen desselben stets unkontrollierbaren, scheinbar zufälligen Schwankungen ausgesetzt bleibt.
</p><p>Diesen Umstand versuchte das folgende Modell zu erklären:
</p><p>Zu Beginn befinden sich in beiden Behältern zusammen eine endliche Anzahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><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"></span> Partikeln; etwa die einzelnen <a href="Molek%C3%BCl" title="Molekül">Moleküle</a> des Stoffes, wovon sich anfangs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{0}\leq N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l_{0}\leq N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3912568a704a1d7c6246a15159e497a55525b980.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.91ex; height:2.509ex;" alt="{\displaystyle l_{0}\leq N}" loading="lazy"></span> im linken und analog <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}=N-l_{0}\leq N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mi>N</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r_{0}=N-l_{0}\leq N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/267d246895c0af69b8ccafc377761201e74f4f37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.015ex; height:2.509ex;" alt="{\displaystyle r_{0}=N-l_{0}\leq N}" loading="lazy"></span> im rechten Behälter aufhalten. In jedem Zeitschritt wird nun genau eines dieser <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><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"></span> Teilchen gleichverteilt ausgewählt, das den Behälter wechselt, sodass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> in jedem Schritt genau um eins ansteigen oder fallen.
</p><p>Mathematisch gesehen handelt es sich bei diesem zufälligen Vorgang um eine <a href="Markow-Kette" title="Markow-Kette">Markow-Kette</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (l_{n})_{n\in \mathbb {N} _{0}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (l_{n})_{n\in \mathbb {N} _{0}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/017553dc42c4e7e2a94e81e5fbed37953d1c65a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.054ex; height:3.009ex;" alt="{\displaystyle (l_{n})_{n\in \mathbb {N} _{0}}}" loading="lazy"></span> mit Zustandsraum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,\ldots N\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mi>N</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\ldots N\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40fb0cb385b339f9ee5b98fec260b2c5a2fcacb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.088ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,\ldots N\}}" loading="lazy"></span> und einer <a href="%C3%9Cbergangsmatrix" title="Übergangsmatrix">Übergangsmatrix</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi =\left(\pi _{a,b}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Π<!-- Π --></mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Pi =\left(\pi _{a,b}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/972aba597bd029d20ecda421dd27df14d3ab2b65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.241ex; height:3.009ex;" alt="{\displaystyle \Pi =\left(\pi _{a,b}\right)}" loading="lazy"></span>, gegeben durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{a,b}={\begin{cases}{\frac {a}{N}},&amp;{\text{falls }}b=a-1\,,\\{\frac {N-a}{N}},&amp;{\text{falls }}b=a+1\,,\\0&amp;{\text{sonst}}.\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>a</mi>
<mi>N</mi>
</mfrac>
</mrow>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls&nbsp;</mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<mi>a</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>N</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
<mi>N</mi>
</mfrac>
</mrow>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls&nbsp;</mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>sonst</mtext>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi _{a,b}={\begin{cases}{\frac {a}{N}},&amp;{\text{falls }}b=a-1\,,\\{\frac {N-a}{N}},&amp;{\text{falls }}b=a+1\,,\\0&amp;{\text{sonst}}.\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80fe21681a0a2b15c9b241b171b7f595515c195c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:31.946ex; height:9.843ex;" alt="{\displaystyle \pi _{a,b}={\begin{cases}{\frac {a}{N}},&amp;{\text{falls }}b=a-1\,,\\{\frac {N-a}{N}},&amp;{\text{falls }}b=a+1\,,\\0&amp;{\text{sonst}}.\end{cases}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Mathematische_Eigenschaften">Mathematische Eigenschaften</h2></div>
<ul><li>Die oben definierte Ehrenfest-Kette besitzt eine eindeutig bestimmte <a href="Station%C3%A4re_Verteilung" title="Stationäre Verteilung">stationäre Verteilung</a>: Ist die Anzahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42239396d24c8ea6a1c13d5015e9958c46e60117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.912ex; height:2.509ex;" alt="{\displaystyle l_{n}}" loading="lazy"></span> der Teilchen im linken (oder rechten) Behälter <a href="Binomialverteilung" title="Binomialverteilung">binomialverteilt</a> mit Parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" loading="lazy"></span>, ist also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(l_{n}=k)={\binom {N}{k}}{\frac {1}{2^{N}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>N</mi>
<mi>k</mi>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(l_{n}=k)={\binom {N}{k}}{\frac {1}{2^{N}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82663a1d01469999bd3ba2fd1152ef811cf51c81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.049ex; height:6.176ex;" alt="{\displaystyle P(l_{n}=k)={\binom {N}{k}}{\frac {1}{2^{N}}}}" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,1,\ldots N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k=0,1,\ldots N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/280e4f007cf2a9142d006a86c53b8aa9eee28c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.876ex; height:2.509ex;" alt="{\displaystyle k=0,1,\ldots N}" loading="lazy"></span>, so hat <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{n+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l_{n+1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca8988cf32e00f927fa922a164e32d24212357b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.012ex; height:2.509ex;" alt="{\displaystyle l_{n+1}}" loading="lazy"></span> dieselbe Verteilung.</li></ul>
<ul><li>Die <a href="Station%C3%A4re_Verteilung#Konvergenz" title="Stationäre Verteilung">Konvergenz</a> der Kette gegen diese Verteilung ist allerdings nicht gegeben, da die Kette <a href="Periodische_Markow-Kette" title="Periodische Markow-Kette">periodisch</a> ist (das erkennt man daran, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42239396d24c8ea6a1c13d5015e9958c46e60117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.912ex; height:2.509ex;" alt="{\displaystyle l_{n}}" loading="lazy"></span> stets zwischen geraden und ungeraden Zahlen wechselt und somit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(l_{n}=k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(l_{n}=k)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40cf0ae7ec0c014bc43af072d7bcc066ec1db57c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.776ex; height:2.843ex;" alt="{\displaystyle P(l_{n}=k)}" loading="lazy"></span> jedes zweite Mal gleich null ist). Dies kann man umgehen, indem man zur <i>aperiodischen Version</i> der Kette übergeht und die Übergangsmatrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Π<!-- Π --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eed3e3db6cc2028a183af948212ed2551d25c954.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 \Pi }" loading="lazy"></span> für einen festen Parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in {]0,1[}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">]</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">[</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p\in {]0,1[}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa00844369e4b0ecff8963fc6b57dcfda8ad3d66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.752ex; height:2.843ex;" alt="{\displaystyle p\in {]0,1[}}" loading="lazy"></span> durch die Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\Pi }}:=pI_{N+1}+(1-p)\Pi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">Π<!-- Π --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>:=</mo>
<mi>p</mi>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mi mathvariant="normal">Π<!-- Π --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\Pi }}:=pI_{N+1}+(1-p)\Pi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c6b7a4060d8b54916d82da9257d8e49002dcf78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.038ex; height:3.343ex;" alt="{\displaystyle {\hat {\Pi }}:=pI_{N+1}+(1-p)\Pi }" loading="lazy"></span> ersetzt (dabei ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{N+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{N+1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8000e9a32fa8dfc530b6558577dd8033b00a9fae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.815ex; height:2.509ex;" alt="{\displaystyle I_{N+1}}" loading="lazy"></span> die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>).<br>Interpretation: mit Wahrscheinlichkeit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> bleibt die Anzahl der Teilchen in den Behältern unverändert, mit Wahrscheinlichkeit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9633a8692121eedfa99cace406205e5d1511ef8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}" loading="lazy"></span> ändert sie sich nach dem oben beschriebenen Verfahren.<br>Dadurch wird die Kette aperiodisch und konvergiert für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1.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\to \infty }" loading="lazy"></span> gegen die stationäre Verteilung, die sich durch diese Modifikation nicht ändert.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>


<p>Gegeben seien zwei Behälter, die durch eine Membran voneinander getrennt sind. In dem linken Behälter befinden sich zu Beginn des Experiments <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>10</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 10}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ec811eb07dcac7ea67b413c5665390a1671ecb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}" loading="lazy"></span> Moleküle und der rechte Behälter ist noch leer. Durch die Membran kann genau ein Molekül pro Zeiteinheit den Behälter wechseln.
</p><p>Da der rechte Behälter zu Beginn noch leer ist, wird in der ersten Sekunde ein Molekül aus dem linken in den rechten Behälter fliegen. Anschließend befinden sich nur noch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>9</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 9}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32d3d1e1f9dfe0254c628379e69a69711fe4eabd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 9}" loading="lazy"></span> Moleküle in dem linken
Behälter. Nun gibt es zwei Möglichkeiten: Entweder eines der verbleibenden <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>9</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 9}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32d3d1e1f9dfe0254c628379e69a69711fe4eabd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 9}" loading="lazy"></span> Moleküle des linken Behälters fliegt in den rechten Bereich, oder das Molekül rechts fliegt wieder zurück in den linken Bereich. Jedes Molekül soll hierbei die gleiche Chance haben, den Behälter zu wechseln. Demnach beträgt die Wahrscheinlichkeit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>90</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 90}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bda6ec8d2b197921d42d52bd57b265cf2c7b392.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 90}" loading="lazy"></span>&nbsp;%, dass ein weiteres Molekül von links nach rechts fliegt. Bei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>8</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 8}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 8}" loading="lazy"></span> Molekülen links beträgt diese Wahrscheinlichkeit nur noch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 80}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>80</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 80}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f60fe9d631c1e752f655cf6fa9c2443198ea9e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 80}" loading="lazy"></span>&nbsp;% und so weiter.
</p><p>Der Übergangsgraph enthält die Zustände <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> bis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>10</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 10}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ec811eb07dcac7ea67b413c5665390a1671ecb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}" loading="lazy"></span>, welche die Anzahl der Moleküle im linken Behälter repräsentieren. Die Markow-Kette startet im Zustand <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>10</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 10}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ec811eb07dcac7ea67b413c5665390a1671ecb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}" loading="lazy"></span>. Vervollständigt man den Übergangsgraphen und erstellt eine dazu passende Übergangsmatrix, kann man die Wahrscheinlichkeitsverteilungen für die Anzahl Moleküle im linken Behälter für jeden Zeitpunkt bestimmen. Nach <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>5</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 5}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29483407999b8763f0ea335cf715a6a5e809f44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}" loading="lazy"></span> Zeiteinheiten besteht mit einer Wahrscheinlichkeit von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30{,}2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>30</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 30{,}2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91732091978b664abbf51f30d9c9e2961909fb6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.134ex; height:2.509ex;" alt="{\displaystyle 30{,}2}" loading="lazy"></span>&nbsp;% zum ersten Mal die Möglichkeit zum physikalischen Gleichgewicht.
</p><p>Die stationäre Verteilung lässt sich mit Hilfe der oben formulierten Formel
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(l_{n}=k)=P(l_{n+1}=k)={\binom {10}{k}}{\frac {1}{2^{10}}}}">
<semantics>
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<p>ermitteln. Dadurch ergibt sich die Wahrscheinlichkeitsverteilung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0.10,0.98,4.39,11.72,20.51,24.61,20.51,11.72,4.39,0.98,0.10)}">
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<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Hans-Otto Georgii: <i>Stochastik: Einführung in die Wahrscheinlichkeitstheorie und Statistik.</i> 4. Auflage. de Gruyter Lehrbuch, Berlin 2009, ISBN 978-3-11-021526-7, S. 166f.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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