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<title>Potts-Modell</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Potts-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>Potts-Modell</b> ist ein <a href="Mathematisches_Modell" title="Mathematisches Modell">mathematisches Modell</a>, welches das in der <a href="Statistische_Physik" title="Statistische Physik">statistischen Physik</a> häufig verwendete <a href="Ising-Modell" title="Ising-Modell">Ising-Modell</a> verallgemeinert. Auf einem Gitter befinden sich statt <a href="Spin" title="Spin">Spins</a> mit nur zwei Zuständen, wie im Ising-Modell, Variablen mit <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 q}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> verschiedenen Zuständen. Im einfachsten Fall beschränkt sich die Wechselwirkung auf benachbarte Gitterplätze.

</p><p>Angewendet wird dieses Modell unter anderem, außer in der statistischen Physik (insbesondere beim Studium von <a href="Phasen%C3%BCbergang" title="Phasenübergang">Phasenübergängen</a>), auch in der <a href="Informatik" title="Informatik">Informatik</a> (<a href="Signalverarbeitung" title="Signalverarbeitung">Signalverarbeitung</a>) und der Biologie (<a href="Neuronales_Netz" title="Neuronales Netz">neuronale Netze</a>). Das Modell wurde nach <a href="Renfrey_Potts" title="Renfrey Potts">Renfrey Potts</a> benannt, welcher das Modell 1951 in seiner Dissertation definierte. Einen Spezialfall behandelten schon 1943 <a href="Julius_Ashkin" title="Julius Ashkin">Julius Ashkin</a> und <a href="Edward_Teller" title="Edward Teller">Edward Teller</a>. Einen Überblick zu Geschichte und Analyse des Modells gibt ein Übersichtsartikel von Fa-Yueh Wu aus dem Jahr 1982.

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

<p>Die Unterscheidung von planarem und Standard-Potts-Modell stammt von <a href="Cyril_Domb" title="Cyril Domb">Cyril Domb</a> (1974)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>.

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<div class="mw-heading mw-heading3"><h3 id="Planares_Potts-Modell">Planares Potts-Modell</h3></div>

<p>Das Potts-Modell besteht aus einem <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 d}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span>-dimensionalen <a href="Gittergraph" title="Gittergraph">Gittergraphen</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(V,E)}">

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<annotation encoding="application/x-tex">{\displaystyle L(V,E)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77d7daaae77bcf88824985a76db33aff6a69fddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.989ex; height:2.843ex;" alt="{\displaystyle L(V,E)}" loading="lazy"></span>, z. B. einem zweidimensionalen Rechteckgitter, einer Menge von Knotenbelegungen <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 S'^{|V|}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc7685a7362553773d89fabc924e0d5935e963e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.385ex; height:2.843ex;" alt="{\displaystyle S'^{|V|}}" loading="lazy"></span> (Knotenkonfigurationen) und einem <a href="Hamiltonoperator" title="Hamiltonoperator">Hamiltonoperator</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 H'}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/575e08b1574dc1c2bb8c5941a2a68d6daca7fd8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.788ex; height:2.509ex;" alt="{\displaystyle H'}" loading="lazy"></span> auf dieser Menge. Jeder Knoten wird belegt mit einem Element aus der Menge

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<annotation encoding="application/x-tex">{\displaystyle S'=\{\theta _{s}={\frac {2\pi s}{q}},0\leq s\leq q\}\quad q\in \{2,3,\ldots \}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/973aa93abc2b25fbd19aebd5ea0d237b8ee949dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.47ex; height:5.676ex;" alt="{\displaystyle S'=\{\theta _{s}={\frac {2\pi s}{q}},0\leq s\leq q\}\quad q\in \{2,3,\ldots \}}" loading="lazy"></span></dd></dl></dd></dl>

<p>Diese können als Punkte auf dem 2-dimensionalen <a href="Einheitskreis" title="Einheitskreis">Einheitskreis</a> interpretiert werden und sind die Richtungen, die die „Spins“ auf den jeweiligen Gitterpunkten annehmen können.

</p><p>Der Hamiltonoperator ist im planaren Potts-Modell (auch Vektor-Potts-Modell oder Uhren-Modell, <i>clock model</i>) gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle H'(\sigma )=J_{1}\sum _{\langle i,j\rangle \in E}\cos(\theta _{s_{i}}-\theta _{s_{j}}),\quad \sigma =(\theta _{s_{1}},\theta _{s_{2}},\dots )\in S'^{|V|}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ce45018dc3f2b26d3ad1ceb395a97d9bb925c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:58.873ex; height:6.009ex;" alt="{\displaystyle H'(\sigma )=J_{1}\sum _{\langle i,j\rangle \in E}\cos(\theta _{s_{i}}-\theta _{s_{j}}),\quad \sigma =(\theta _{s_{1}},\theta _{s_{2}},\dots )\in S'^{|V|}.}" loading="lazy"></span></dd></dl>

<p>Summiert wird über alle benachbarten Knoten <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 \langle i,j\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle i,j\rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e4fc4ffd5517cd57452a53c4617698157adb631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle \langle i,j\rangle }" 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 i,j\in V}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cdc20c53e2e4e6cee5bb2c4d52bb00f5bdc079e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.422ex; height:2.509ex;" alt="{\displaystyle i,j\in V}" loading="lazy"></span>. Die <a href="Kopplungskonstante" title="Kopplungskonstante">Kopplungskonstante</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 J_{1}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/260ffe7da7c858cf114ad89a6c794944ea4e760f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{1}}" loading="lazy"></span> beschreibt die Wechselwirkung zwischen Spins auf den benachbarten Knoten.

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<div class="mw-heading mw-heading3"><h3 id="Standard-Potts-Modell">Standard-Potts-Modell</h3></div>

<p>Alternativ zum gerade beschriebenen planaren Potts-Modell gibt es das Standard-Potts-Modell (oder einfach: Potts-Modell). Dabei werden die Knoten belegt mit Elementen aus der Menge

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d18ef97e2819dd805bdbd6d6607883561b0242d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.529ex; height:2.843ex;" alt="{\displaystyle S=\{0,1,\dots ,q\}}" loading="lazy"></span>.</dd></dl></dd></dl>

<p>Der Hamiltonoperator ist hier gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\sum _{\langle i,j\rangle \in E}\delta (s_{i},s_{j}),\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|},}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd3537225e955d1bb632f7bf9a33a7966e1245fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:51.565ex; height:6.009ex;" alt="{\displaystyle H(\sigma )=-J\sum _{\langle i,j\rangle \in E}\delta (s_{i},s_{j}),\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|},}" loading="lazy"></span></dd></dl>

<p>wobei <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 \delta }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span> das <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a> ist.

</p><p>Das heißt, falls zwei benachbarte Knoten verschiedene Werte der Spins besitzen, verschwindet der entsprechende Summand. Das negative Vorzeichen 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 J}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> ist eine Konvention, motiviert vom Ising-Modell. Das Standard-Potts-Modell ist <a href="Ferromagnetisch" class="mw-redirect" title="Ferromagnetisch">ferromagnetisch</a> 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 J>0}">

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<annotation encoding="application/x-tex">{\displaystyle J>0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f57295e4ce96b4d40ef57c8e53f56380eadb31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.732ex; height:2.176ex;" alt="{\displaystyle J>0}" loading="lazy"></span> und <a href="Antiferromagnetisch" class="mw-redirect" title="Antiferromagnetisch">antiferromagnetisch</a> 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 J<0}">

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<annotation encoding="application/x-tex">{\displaystyle J<0}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Verhältnis_zu_anderen_statistischen_Modellen"><span id="Verh.C3.A4ltnis_zu_anderen_statistischen_Modellen"></span>Verhältnis zu anderen statistischen Modellen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Allgemeine_Version">Allgemeine Version</h3></div>

<p>Auf dem Gittergraphen <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(V,E)}">

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<annotation encoding="application/x-tex">{\displaystyle L(V,E)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77d7daaae77bcf88824985a76db33aff6a69fddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.989ex; height:2.843ex;" alt="{\displaystyle L(V,E)}" loading="lazy"></span> mit der Menge der Knotenbelegungen <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 \{1,2,\dots ,q-1\}^{|V|}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,2,\dots ,q-1\}^{|V|}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92471e0b44f753fd572b952fa634e8b89f3c3345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.345ex; height:3.343ex;" alt="{\displaystyle \{1,2,\dots ,q-1\}^{|V|}}" loading="lazy"></span> kann eine allgemeinere Version des Potts-Modells definiert werden:

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-\sum _{\langle i,j\rangle \in E}J_{ij}\delta (s_{i},s_{j})-\beta ^{-1}\sum _{i\in \{1,2,\dots \,|V|\}}h_{i}s_{i},\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5254e265df226758dd3aa7bf83d3c0dd339fab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:74.532ex; height:6.009ex;" alt="{\displaystyle H(\sigma )=-\sum _{\langle i,j\rangle \in E}J_{ij}\delta (s_{i},s_{j})-\beta ^{-1}\sum _{i\in \{1,2,\dots \,|V|\}}h_{i}s_{i},\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|}.}" loading="lazy"></span></dd></dl>

<p>Im Unterschied zum ursprünglichen Modell variiert die Wechselwirkung zwischen den benachbarten Knoten. Außerdem kann ein äußeres Feld ergänzt werden:

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<annotation encoding="application/x-tex">{\displaystyle -\beta ^{-1}\sum _{i\in \{1,2,\dots ,|V|\}}h_{i}s_{i}}</annotation>

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<p>Hierbei ist wie üblich <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 \beta ={\tfrac {1}{k_{\mathrm {B} }T}}}">

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<annotation encoding="application/x-tex">{\displaystyle \beta ={\tfrac {1}{k_{\mathrm {B} }T}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa7f02ed2dad1b921ea6bfe65971fc756eed141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.389ex; height:4.009ex;" alt="{\displaystyle \beta ={\tfrac {1}{k_{\mathrm {B} }T}}}" loading="lazy"></span> mit der <a href="Boltzmann-Konstante" title="Boltzmann-Konstante">Boltzmann-Konstanten</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 k_{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\mathrm {B} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\mathrm {B} }}" loading="lazy"></span> und der <a href="Absolute_Temperatur" class="mw-redirect" title="Absolute Temperatur">Temperatur</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 T}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span>.

</p><p>Die Wechselwirkungen <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 J_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a5daff3ca4e673277d8780ceb49b6922bbf6fac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.767ex; height:2.843ex;" alt="{\displaystyle J_{ij}}" loading="lazy"></span> müssen nicht auf nächstbenachbarte Gitterplätze beschränkt werden. In <i>verdünnten Potts-Modellen</i> gibt es freie Gitterplätze (<i>Gitter-Gas</i>) oder auch Wechselwirkungen verschiedener Stärke. Durch geeignete <a href="Randbedingungen" class="mw-redirect" title="Randbedingungen">Randbedingungen</a> können interessante Effekte, wie z. B. <a href="Benetzung" title="Benetzung">Benetzung</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> oder Grenzflächen<a href="Adsorption" title="Adsorption">adsorption</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>, induziert werden.

</p>

<div class="mw-heading mw-heading3"><h3 id="Das_Ising-Modell_als_Spezialfall">Das Ising-Modell als Spezialfall</h3></div>

<p>Setzt man <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 q=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26622af6012fb982cab4e9584f57dd4f364233b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=2}" loading="lazy"></span>, so folgt aus dem Potts-Modell das <a href="Ising-Modell" title="Ising-Modell">Ising-Modell</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Das_XY-Modell_als_Spezialfall">Das XY-Modell als Spezialfall</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953a2f08f9d9ab42fcec5c6afeea8cb551e42c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.007ex; height:2.176ex;" alt="{\displaystyle q\to \infty }" loading="lazy"></span> erhält man das <a href="XY-Modell" title="XY-Modell">XY-Modell</a>, welches wiederum als Spezialfall des <a href="N-Vektor-Modell" title="N-Vektor-Modell">N-Vektor-Modells</a> mit <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=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/405d64b14536deffc3465f1e81b1b7fe9358ad2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=2}" loading="lazy"></span> verstanden werden kann. Betrachtet man das planare Potts-Modell, so ist der <a href="Zustandsraum" class="mw-disambig" title="Zustandsraum">Zustandsraum</a> der Spins keine endliche <a href="Teilmenge" title="Teilmenge">Teilmenge</a> des Einheitskreises, sondern der <i>ganze</i> 2-dimensionale Einheitskreis.

</p>

<div class="mw-heading mw-heading3"><h3 id="Ashkin-Teller-Modell">Ashkin-Teller-Modell</h3></div>

<p>Das Ashkin-Teller-Modell ist das planare Potts-Modell mit <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 q=4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed969e29cea24ad34d904c3574fade94cbfca08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=4}" loading="lazy"></span> Zuständen.

</p>

<div class="mw-heading mw-heading3"><h3 id="Sonstige">Sonstige</h3></div>

<p>Es gibt auch Verbindungen zum <a href="Heisenberg-Modell" title="Heisenberg-Modell">Heisenberg-Modell</a>, N-Vektor-Modell, (ice-rule-)Vertex-Modellen und zur <a href="Perkolationstheorie" title="Perkolationstheorie">Perkolationstheorie</a> (zuerst von P.W. Kasteleyn und C.M. Fortuin 1969<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> für Bond-Perkolation, später auch für Site-Perkolation).

</p><p>Das <a href="Kirchhoffsche_Regeln" title="Kirchhoffsche Regeln">Kirchhoffsche Gesetz</a> für Netzwerke aus linearen Widerständen ergibt sich als <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 q=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/654c2d5dc1a26e0af36dc0deb5fd252c6178977a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=0}" loading="lazy"></span> Grenzwert des Potts-Modells (Kasteleyn, Fortuin 1972).

</p>

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

<p>Potts betrachtete das planare Modell und konnte ähnlich wie beim Ising-Modell mit Kramers-Wannier-Dualität den kritischen Punkt bestimmen für das Rechteckgitter 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 q=2,3,4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=2,3,4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f45e65c9fc438bc91a157c908434019eee97c39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.723ex; height:2.509ex;" alt="{\displaystyle q=2,3,4}" loading="lazy"></span>. Am Ende seiner Arbeit gab er den kritischen Punkt des Standard-Potts-Modells für alle <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 q}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>.

</p><p>Das planare und das Standard-Modell sind identisch 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 q=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26622af6012fb982cab4e9584f57dd4f364233b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=2}" loading="lazy"></span> (Ising-Modell) mit <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 J=2J_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J=2J_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b2b29835b57784e66ff40c29e6178d038bad1d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.077ex; height:2.509ex;" alt="{\displaystyle J=2J_{1}}" loading="lazy"></span> und 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 q=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=3}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d859ff2e5023ed1f714ccace69e88ab993a5f43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=3}" loading="lazy"></span> mit <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 J={\tfrac {3}{2}}J_{1}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J={\tfrac {3}{2}}J_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c287ba8711c7db7ee3f378ef57b4e920334d9a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.572ex; height:3.509ex;" alt="{\displaystyle J={\tfrac {3}{2}}J_{1}}" loading="lazy"></span>. Außerdem ist das planare Modell mit <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 q=4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed969e29cea24ad34d904c3574fade94cbfca08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=4}" loading="lazy"></span> für beliebige Gitter auf das Modell mit <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 q=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26622af6012fb982cab4e9584f57dd4f364233b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=2}" loading="lazy"></span> <a href="Reduzierbar" class="mw-redirect mw-disambig" title="Reduzierbar">reduzierbar</a>. 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 q>4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q>4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af8d89f406bd58d810ef450981499e050993c9fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q>4}" loading="lazy"></span> gibt es dagegen keine offensichtlichen Bezüge zwischen dem planaren und dem Standard-Modell.

</p><p>Auf einem zweidimensionalen Gitter hat das Potts-Modell mit <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 J>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J>0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f57295e4ce96b4d40ef57c8e53f56380eadb31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.732ex; height:2.176ex;" alt="{\displaystyle J>0}" loading="lazy"></span> einen <a href="Phasen%C3%BCbergang#Klassifikation_nach_Ehrenfest" title="Phasenübergang">Phasenübergang erster Ordnung</a> 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 q>4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q>4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af8d89f406bd58d810ef450981499e050993c9fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q>4}" loading="lazy"></span> und ansonsten einen kontinuierlicher Phasenübergang (2. Ordnung) wie beim Isingmodell (<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 q=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26622af6012fb982cab4e9584f57dd4f364233b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=2}" loading="lazy"></span>) (<a href="Rodney_Baxter" title="Rodney Baxter">Rodney Baxter</a> 1973, 1978). Baxter benutzte dabei die Identifizierung des zweidimensionalen Potts-Modells mit dem Ice-rule-Vertexmodell durch Temperley und <a href="Elliott_Lieb" title="Elliott Lieb">Elliott Lieb</a> (1971 für ein Gitter aus Quadraten).

</p><p>Das eindimensionale Potts-Modell ist exakt lösbar (mit Hilfe der Transfer-Matrix-Methode) und ebenso das zweidimensionale Modell mit Wechselwirkungen zwischen benachbarten Gitterplätzen. Im Allgemeinen liefern insbesondere <a href="Monte-Carlo-Simulation" title="Monte-Carlo-Simulation">Monte-Carlo-Simulationen</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> und die <a href="Renormierungsgruppe" title="Renormierungsgruppe">Renormierungsgruppentheorie</a> verlässliche Ergebnisse.

</p>

<div class="mw-heading mw-heading2"><h2 id="Potts-Maß"><span id="Potts-Ma.C3.9F"></span>Potts-Maß</h2></div>

<p>Mit der <a href="Hamilton-Funktion" title="Hamilton-Funktion">Hamilton-Funktion</a> wie oben

</p>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\sum _{\langle i,j\rangle \in E}\delta (s_{i},s_{j}),\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9df7c3667a73f4302ef94ae3e1861eef6c00467b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:51.565ex; height:6.009ex;" alt="{\displaystyle H(\sigma )=-J\sum _{\langle i,j\rangle \in E}\delta (s_{i},s_{j}),\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|}.}" loading="lazy"></span></dd></dl>

<p>und der üblichen Definition der <a href="Zustandssumme" title="Zustandssumme">Zustandssumme</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 Z}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span>

</p>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\sum _{\sigma \in S^{|V|}}e^{-\beta H(\sigma )},\quad \beta \in {]0,\infty [}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b733b69d4ead47b5b58823ea0611d957f58cf43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:34.557ex; height:6.176ex;" alt="{\displaystyle Z(\beta )=\sum _{\sigma \in S^{|V|}}e^{-\beta H(\sigma )},\quad \beta \in {]0,\infty [}.}" loading="lazy"></span></dd></dl>

<p>kann man das Potts-Maß <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 }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span> definieren, das als <a href="Wahrscheinlichkeitsma%C3%9F" title="Wahrscheinlichkeitsmaß">Wahrscheinlichkeitsmaß</a> zu den <a href="Boltzmann-Statistik" title="Boltzmann-Statistik">Boltzmannverteilungen</a> gehört:

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (\sigma )=Z(\beta )^{-1}e^{-\beta H(\sigma )},\quad \sigma \in S^{|V|}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d490eb14db2874836dbb368af3863058ff294c62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.045ex; height:3.343ex;" alt="{\displaystyle \pi (\sigma )=Z(\beta )^{-1}e^{-\beta H(\sigma )},\quad \sigma \in S^{|V|}.}" loading="lazy"></span></dd></dl>

<p>Die <a href="Freie_Energie" title="Freie Energie">freie Energie</a> ist wie üblich:

</p>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle F(\beta )=-\beta ^{-1}\ln(Z(\beta )),\quad \beta \in {]0,\infty [}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f39ef5ee5cb169c73fa8f0ec23bcbbe8eed22009.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.406ex; height:3.176ex;" alt="{\displaystyle F(\beta )=-\beta ^{-1}\ln(Z(\beta )),\quad \beta \in {]0,\infty [}.}" loading="lazy"></span></dd></dl>

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

<ul><li>Julius Ashkin, Edward Teller: <cite style="font-style:italic">Statistics of Two-Dimensional Lattices With Four Components</cite>. In: <cite style="font-style:italic"><a href="Physical_Review" title="Physical Review">Phys. Rev.</a></cite> 64. Jahrgang, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>5–6</span>, 1943, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>178–184</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1103/PhysRev.64.178">10.1103/PhysRev.64.178</a></span>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1943PhRv...64..178A">1943PhRv...64..178A</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:Potts-Modell&rft.atitle=Statistics+of+Two-Dimensional+Lattices+With+Four+Components&rft.au=Julius%26%2332%3BAshkin%2C%26%2332%3BEdward%26%2332%3BTeller&rft.date=1943&rft.doi=10.1103%2FPhysRev.64.178&rft.genre=journal&rft.issue=5-6&rft.jtitle=Phys.+Rev.&rft.pages=178-184&rft.volume=64.+Jahrgang" style="display:none"> </span></li>

<li>Renfrey B. Potts: <cite style="font-style:italic">Some Generalized Order–Disorder Transformations</cite>. In: <cite style="font-style:italic">Mathematical Proceedings of the Cambridge Philosophical Society</cite>. 48. Jahrgang, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, 1952, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>106–109</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1017/S0305004100027419">10.1017/S0305004100027419</a></span>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952PCPS...48..106P">1952PCPS...48..106P</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:Potts-Modell&rft.atitle=Some+Generalized+Order-Disorder+Transformations&rft.au=Renfrey+B.%26%2332%3BPotts&rft.date=1952&rft.doi=10.1017%2FS0305004100027419&rft.genre=journal&rft.issue=1&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.pages=106-109&rft.volume=48.+Jahrgang" style="display:none"> </span></li>

<li>Fa-Yueh Wu: <cite style="font-style:italic">The Potts model</cite>. In: <cite style="font-style:italic"><a href="Reviews_of_Modern_Physics" class="mw-redirect" title="Reviews of Modern Physics">Reviews of Modern Physics</a></cite>. 54. Jahrgang, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, 1982, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>235–268</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1103/RevModPhys.54.235">10.1103/RevModPhys.54.235</a></span>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982RvMP...54..235W">1982RvMP...54..235W</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:Potts-Modell&rft.atitle=The+Potts+model&rft.au=Fa-Yueh%26%2332%3BWu&rft.date=1982&rft.doi=10.1103%2FRevModPhys.54.235&rft.genre=journal&rft.issue=1&rft.jtitle=Reviews+of+Modern+Physics&rft.pages=235-268&rft.volume=54.+Jahrgang" style="display:none"> </span></li></ul>

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

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="Siegfried_Dietrich_(Physiker)" title="Siegfried Dietrich (Physiker)">S.Dietrich</a>, in <i>Phase transitions and critical phenomena</i> (Hrsg. <a href="Cyril_Domb" title="Cyril Domb">C. Domb</a> und <a href="Joel_Lebowitz" title="Joel Lebowitz">J.L. Lebowitz</a>), Band 12, 1988 </span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a href="Walter_Selke" title="Walter Selke">W.Selke</a>, <a href="Werner_Pesch" title="Werner Pesch">W.Pesch</a>, Z. Phys. B, Band 47, 1982, S. 335</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a href="Pieter_Kasteleyn" title="Pieter Kasteleyn">P. W. Kasteleyn</a>, C.M. Fortuin,

J. Phys. Soc. Japan, Band 26 (Suppl.),1969, S. 11</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><a href="David_P._Landau" title="David P. Landau">D.P.Landau</a>, <a href="Kurt_Binder" title="Kurt Binder">K.Binder</a>, <i>A Guide to Monte Carlo Simulations in Statistical Physics</i>, 2014 </span>

</li>

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