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Ising model

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content plainlist">

<ul><li><a href="Thermodynamics" title="Thermodynamics">Thermodynamics</a></li>

<li><a href="Kinetic_theory_of_gases" title="Kinetic theory of gases">Kinetic theory</a></li></ul></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="Particle_statistics" title="Particle statistics">Particle statistics</a></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">Spin–statistics theorem</a></li>

<li><a href="Indistinguishable_particles" title="Indistinguishable particles">Indistinguishable particles</a></li>

<li><a href="Maxwell%E2%80%93Boltzmann_statistics" title="Maxwell–Boltzmann statistics">Maxwell–Boltzmann</a></li>

<li><a href="Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein</a></li>

<li><a href="Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac</a></li>

<li><a href="Parastatistics" title="Parastatistics">Parastatistics</a></li>

<li><a href="Anyon" title="Anyon">Anyonic statistics</a></li>

<li><a href="Braid_statistics" title="Braid statistics">Braid statistics</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">Thermodynamic ensembles</a></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li>NVE <a href="Microcanonical_ensemble" title="Microcanonical ensemble">Microcanonical</a></li>

<li>NVT <a href="Canonical_ensemble" title="Canonical ensemble">Canonical</a></li>

<li>µVT <a href="Grand_canonical_ensemble" title="Grand canonical ensemble">Grand canonical</a></li>

<li>NPH <a href="Isoenthalpic%E2%80%93isobaric_ensemble" title="Isoenthalpic–isobaric ensemble">Isoenthalpic–isobaric</a></li>

<li>NPT <a href="Isothermal%E2%80%93isobaric_ensemble" title="Isothermal–isobaric ensemble">Isothermal–isobaric</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Models</div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="Debye_model" title="Debye model">Debye</a></li>

<li><a href="Einstein_solid" title="Einstein solid">Einstein</a></li>

<li><a href="Potts_model" title="Potts model">Potts</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="Internal_energy" title="Internal energy">Internal energy</a></li>

<li><a href="Enthalpy" title="Enthalpy">Enthalpy</a></li>

<li><a href="Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a></li>

<li><a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a></li>

<li><a href="Grand_potential" title="Grand potential">Grand potential / Landau free energy</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>

<li><a href="Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li>

<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li>

<li><a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Paul_Dirac" title="Paul Dirac">Dirac</a></li>

<li><a href="Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li>

<li><a href="Richard_C._Tolman" title="Richard C. Tolman">Tolman</a></li>

<li><a href="Peter_Debye" title="Peter Debye">Debye</a></li>

<li><a href="Enrico_Fermi" title="Enrico Fermi">Fermi</a></li>

<li><a href="John_Lighton_Synge" title="John Lighton Synge">Synge</a></li>

<li><a href="Ernst_Ising" title="Ernst Ising">Ising</a></li>

<li><a href="Lev_Landau" title="Lev Landau">Landau</a></li></ul></div></div></td>

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The Ising model (or Lenz–Ising model), named after the physicists <a href="Ernst_Ising" title="Ernst Ising">Ernst Ising</a> and <a href="Wilhelm_Lenz" title="Wilhelm Lenz">Wilhelm Lenz</a>, is a <a href="Mathematical_models_in_physics" class="mw-redirect" title="Mathematical models in physics">mathematical model</a> of <a href="Ferromagnetism" title="Ferromagnetism">ferromagnetism</a> in <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. The model consists of <a href="Discrete_variables" class="mw-redirect" title="Discrete variables">discrete variables</a> that represent <a href="Nuclear_magnetic_moment" title="Nuclear magnetic moment">magnetic dipole moments of atomic "spins"</a> that can be in one of two states (+1 or −1). The spins are arranged in a <a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">graph</a>, usually a <a href="Lattice_(group)" title="Lattice (group)">lattice</a> (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The two-dimensional <a href="Square-lattice_Ising_model" class="mw-redirect" title="Square-lattice Ising model">square-lattice Ising model</a> is one of the simplest statistical models to show a <a href="Phase_transition" title="Phase transition">phase transition</a>.<a href="#cite_note-1">[1]</a> Though it is a highly simplified model of a magnetic material, the Ising model can still provide qualitative and sometimes quantitative results applicable to real physical systems.

The Ising model was invented by the physicist <a href="Wilhelm_Lenz" title="Wilhelm Lenz">Wilhelm Lenz</a> (<a href="#CITEREFLenz1920">1920</a>), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by <a href="#CITEREFIsing1925">Ising (1925)</a> alone in his 1924 thesis;<a href="#cite_note-2">[2]</a> it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by <a href="Lars_Onsager" title="Lars Onsager">Lars Onsager</a> (<a href="#CITEREFOnsager1944">1944</a>). It is usually solved by a <a href="Transfer-matrix_method_(statistical_mechanics)" title="Transfer-matrix method (statistical mechanics)">transfer-matrix method</a>, although there exists a very simple approach relating the model to a non-interacting fermionic <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>.<a href="#cite_note-3">[3]</a>

In dimensions greater than four, the phase transition of the Ising model is described by <a href="Mean-field_theory" title="Mean-field theory">mean-field theory</a>. The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970s, culminating in an exact solution of the zero-field, time-independent <a href="#CITEREFBarth1981">Barth (1981)</a> model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible <a class="mw-selflink-fragment" href="#Applications">applications</a>.

The Ising problem without an external field can be equivalently formulated as a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> <a href="Maximum_cut" title="Maximum cut">maximum cut</a> (Max-Cut) problem that can be solved via <a href="Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a>.

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

Consider a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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

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</math><img src="./0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> of lattice sites, each with a set of adjacent sites (e.g. a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>) forming a <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><img src="./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">-dimensional lattice. For each lattice site <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \Lambda }">

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

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</math><img src="./63f2ff50e818a1bb3c26de3f31dbf21023958692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.665ex; height:2.176ex;" alt="{\displaystyle k\in \Lambda }" loading="lazy"> there is a discrete variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}}">

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

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</math><img src="./330415aef0a06818b843c55814fda645b793811b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.416ex; height:2.009ex;" alt="{\displaystyle \sigma _{k}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}\in \{-1,+1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{k}\in \{-1,+1\}}</annotation>

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</math><img src="./97ee7f84daa05e850b60bf5fdf4c712696f4a370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.557ex; height:2.843ex;" alt="{\displaystyle \sigma _{k}\in \{-1,+1\}}" loading="lazy">, representing the site's spin. A spin configuration, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sigma }=\{\sigma _{k}\}_{k\in \Lambda }}">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\sigma }=\{\sigma _{k}\}_{k\in \Lambda }}</annotation>

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</math><img src="./194c5e07a92045371cf4b7a5d9a792af8c1ba9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.495ex; height:2.843ex;" alt="{\displaystyle {\sigma }=\{\sigma _{k}\}_{k\in \Lambda }}" loading="lazy"> is an assignment of spin value to each lattice site.

For any two adjacent sites <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in \Lambda }">

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

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</math><img src="./ea239edebaeb9ac27281f61cc5491fea50c2fd4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.248ex; height:2.509ex;" alt="{\displaystyle i,j\in \Lambda }" loading="lazy"> there is an interaction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}}">

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

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</math><img src="./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">. Also a site <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in \Lambda }">

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

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</math><img src="./8a1f02a9c168b3003d865f1890ec0017f8082e2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.438ex; height:2.509ex;" alt="{\displaystyle j\in \Lambda }" loading="lazy"> has an external magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{j}}">

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

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</math><img src="./614689f15f73ad5b4a5d7fa837a72614202b0d89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.249ex; height:2.843ex;" alt="{\displaystyle h_{j}}" loading="lazy"> interacting with it. The energy of a configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sigma }}">

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

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</math><img src="./314ac5cfd38ebefe34513c781f56fa6510fa4fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {\sigma }}" loading="lazy"> is given by the <a href="Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonian function</a>

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-\sum _{\langle ij\rangle }J_{ij}\sigma _{i}\sigma _{j}-\mu \sum _{j}h_{j}\sigma _{j},}</annotation>

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where the first sum is over pairs of adjacent spins (every pair is counted once). The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle ij\rangle }">

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

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</math><img src="./4f5b644a2ce15105abd60c833e67c30ca4960800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.57ex; height:2.843ex;" alt="{\displaystyle \langle ij\rangle }" loading="lazy"> indicates that sites <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> are nearest neighbors. The <a href="Magnetic_moment" title="Magnetic moment">magnetic moment</a> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">. Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally.<a href="#cite_note-4">[4]</a> The Ising Hamiltonian is an example of a <a href="Pseudo-Boolean_function" title="Pseudo-Boolean function">pseudo-Boolean function</a>; tools from the <a href="Analysis_of_Boolean_functions" title="Analysis of Boolean functions">analysis of Boolean functions</a> can be applied to describe and study it.

The configuration probability is given by the <a href="Boltzmann_distribution" title="Boltzmann distribution">Boltzmann distribution</a> with <a href="Inverse_temperature" class="mw-redirect" title="Inverse temperature">inverse temperature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \geq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./78f6fcdda6c164f2e0c8df177d9917a5d0c94214.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta \geq 0}" loading="lazy">:

<msub>

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<mfrac>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\beta }(\sigma )={\frac {e^{-\beta H(\sigma )}}{Z_{\beta }}},}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =1/(k_{\text{B}}T)}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d040576e6814f3f5dffdead8f366a891566c3215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.808ex; height:2.843ex;" alt="{\displaystyle \beta =1/(k_{\text{B}}T)}" loading="lazy">, and the normalization constant

<msub>

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</mrow>

</munder>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{\beta }=\sum _{\sigma }e^{-\beta H(\sigma )}}</annotation>

</semantics>

</math>

is the <a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a>. For a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> of the spins ("observable"), one denotes by

<msub>

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</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle f\rangle _{\beta }=\sum _{\sigma }f(\sigma )P_{\beta }(\sigma )}</annotation>

</semantics>

</math>

the expectation (mean) value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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">.

The configuration probabilities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\beta }(\sigma )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\beta }(\sigma )}</annotation>

</semantics>

</math><img src="./45b3a57760ea6c60313ed1b7246239644c3f2ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.805ex; height:3.009ex;" alt="{\displaystyle P_{\beta }(\sigma )}" loading="lazy"> represent the probability that (in equilibrium) the system is in a state with configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">.

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

The minus sign on each term of the Hamiltonian function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\sigma )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0617cdf63ce0b02a28bbb6b67829da8259a69045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle H(\sigma )}" loading="lazy"> is conventional. Using this sign convention, Ising models can be classified according to the sign of the interaction: if, for a pair i, j

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3a8820027907e8172e8979cb8075635ddf29d418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.028ex; height:2.843ex;" alt="{\displaystyle J_{ij}>0}" loading="lazy">, the interaction is called <a href="Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a>,</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}<0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./72c1d9cfa95e725ab1ebb596bc2f2f99a95a23e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.028ex; height:2.843ex;" alt="{\displaystyle J_{ij}<0}" loading="lazy">, the interaction is called <a href="Antiferromagnetic" class="mw-redirect" title="Antiferromagnetic">antiferromagnetic</a>,</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6107d586d9b443fd27e55fa1dfd296b5d820256a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.028ex; height:2.843ex;" alt="{\displaystyle J_{ij}=0}" loading="lazy">, the spins are noninteracting.</li></ul></div>

The system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic. The original Ising models were ferromagnetic, and it is still often assumed that "Ising model" means a ferromagnetic Ising model.

In a ferromagnetic Ising model, spins desire to be aligned: the configurations in which adjacent spins are of the same sign have higher probability. In an antiferromagnetic model, adjacent spins tend to have opposite signs.

The sign convention of H(σ) also explains how a spin site j interacts with the external field. Namely, the spin site wants to line up with the external field. If:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./00333f3e95acec7b318d7a3cf9701a57b6bd6d56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle h_{j}>0}" loading="lazy">, the spin site j desires to line up in the positive direction,</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{j}<0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./25648d556d208e83e1def558eaa313b45aefb321.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle h_{j}<0}" loading="lazy">, the spin site j desires to line up in the negative direction,</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{j}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./44e120fa16aae13d0f20dd36af71581fa6b7195a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle h_{j}=0}" loading="lazy">, there is no external influence on the spin site.</li></ul></div>

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

Ising models are often examined without an external field interacting with the lattice, that is, h = 0 for all j in the lattice Λ. Using this simplification, the Hamiltonian becomes

</mrow>

</munder>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-\sum _{\langle i~j\rangle }J_{ij}\sigma _{i}\sigma _{j}.}</annotation>

</semantics>

</math>

When the external field is zero everywhere, h = 0, the Ising model is symmetric under switching the value of the spin in all the lattice sites; a nonzero field breaks this symmetry.

Another common simplification is to assume that all of the nearest neighbors ⟨ij⟩ have the same interaction strength. Then we can set Jij = J for all pairs i, j in Λ. In this case the Hamiltonian is further simplified to

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\sum _{\langle i~j\rangle }\sigma _{i}\sigma _{j}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Connection_to_graph_maximum_cut">Connection to <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> <a href="Maximum_cut" title="Maximum cut">maximum cut</a></h3></div>

A subset S of the <a href="Vertex_(graph_theory)" title="Vertex (graph theory)">vertex</a> set V(G) of a weighted undirected graph G determines a cut of the graph G into S and its <a href="Complement_graph" title="Complement graph">complementary</a> subset G\S. The size of the cut is the sum of the weights of the edges between S and G\S. A <a href="Maximum_cut" title="Maximum cut">maximum cut</a> size is at least the size of any other cut, varying S.

For the Ising model without an external field on a graph G, the Hamiltonian becomes the following sum over the graph edges E(G)

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-\sum _{ij\in E(G)}J_{ij}\sigma _{i}\sigma _{j}}</annotation>

</semantics>

</math><img src="./106451dd3bd9bf7d97dad5896889cf9d6bc78d55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.183ex; height:6.009ex;" alt="{\displaystyle H(\sigma )=-\sum _{ij\in E(G)}J_{ij}\sigma _{i}\sigma _{j}}" loading="lazy">.</dd></dl>

Here each vertex i of the graph is a spin site that takes a spin value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}=\pm 1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b113c6e8afd38cbb80a9c0d852ce8aaed4892577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.196ex; height:2.509ex;" alt="{\displaystyle \sigma _{i}=\pm 1}" loading="lazy">. A given spin configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> partitions the set of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(G)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8ba39dee5fd7f4467e387af4026315fb1fb21628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle V(G)}" loading="lazy"> into two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-depended subsets, those with spin up <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b2aafc830b7719ab820d87c8a8eec52e8756b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle V^{+}}" loading="lazy"> and those with spin down <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{-}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fa9ed53745ca8c9a03fbbb6c28c341c067aaa0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle V^{-}}" loading="lazy">. We denote by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (V^{+})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (V^{+})}</annotation>

</semantics>

</math><img src="./bec2405fa62c2e32a8f2bd285d3f989088e4df1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.286ex; height:3.009ex;" alt="{\displaystyle \delta (V^{+})}" loading="lazy"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-depended set of edges that connects the two complementary vertex subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b2aafc830b7719ab820d87c8a8eec52e8756b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle V^{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{-}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fa9ed53745ca8c9a03fbbb6c28c341c067aaa0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle V^{-}}" loading="lazy">. The size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\delta (V^{+})\right|}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|\delta (V^{+})\right|}</annotation>

</semantics>

</math><img src="./2888bc4d5cbed56e4e6b361881154159e7de5e87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.579ex; height:3.176ex;" alt="{\displaystyle \left|\delta (V^{+})\right|}" loading="lazy"> of the cut <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (V^{+})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (V^{+})}</annotation>

</semantics>

</math><img src="./bec2405fa62c2e32a8f2bd285d3f989088e4df1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.286ex; height:3.009ex;" alt="{\displaystyle \delta (V^{+})}" loading="lazy"> to <a href="Bipartite_graph" title="Bipartite graph">bipartite</a> the weighted undirected graph G can be defined as

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|\delta (V^{+})\right|={\frac {1}{2}}\sum _{ij\in \delta (V^{+})}W_{ij},}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./29c09e9d719bb634d8ca5a6172b0562b945bf325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.671ex; height:2.843ex;" alt="{\displaystyle W_{ij}}" loading="lazy"> denotes a weight of the edge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ij}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./53fcc7b57da64979c370eb150eb5a61a625a08e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.761ex; height:2.509ex;" alt="{\displaystyle ij}" loading="lazy"> and the scaling 1/2 is introduced to compensate for double counting the same weights <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{ij}=W_{ji}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8a0e4bdaadbec6dece52add45f8ac9b79e9aa59b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.44ex; height:2.843ex;" alt="{\displaystyle W_{ij}=W_{ji}}" loading="lazy">.

The identities

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H(\sigma )&=-\sum _{ij\in E(V^{+})}J_{ij}-\sum _{ij\in E(V^{-})}J_{ij}+\sum _{ij\in \delta (V^{+})}J_{ij}\\&=-\sum _{ij\in E(G)}J_{ij}+2\sum _{ij\in \delta (V^{+})}J_{ij},\end{aligned}}}</annotation>

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where the total sum in the first term does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">, imply that minimizing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\sigma )}">

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

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</math><img src="./0617cdf63ce0b02a28bbb6b67829da8259a69045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle H(\sigma )}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is equivalent to minimizing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{ij\in \delta (V^{+})}J_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{ij\in \delta (V^{+})}J_{ij}}</annotation>

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</math><img src="./b0f060abc80f86f4cdecaff82eee0dac7366a16e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:10.074ex; height:6.176ex;" alt="{\displaystyle \sum _{ij\in \delta (V^{+})}J_{ij}}" loading="lazy">. Defining the edge weight <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{ij}=-J_{ij}}">

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

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</math><img src="./f3c44147f7dbc01e722461eda8c94dfd5a7d46cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.345ex; height:2.843ex;" alt="{\displaystyle W_{ij}=-J_{ij}}" loading="lazy"> thus turns the Ising problem without an external field into a graph Max-Cut problem

<a href="#cite_note-:0-5">[5]</a> maximizing the cut size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\delta (V^{+})\right|}">

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<annotation encoding="application/x-tex">{\displaystyle \left|\delta (V^{+})\right|}</annotation>

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</math><img src="./2888bc4d5cbed56e4e6b361881154159e7de5e87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.579ex; height:3.176ex;" alt="{\displaystyle \left|\delta (V^{+})\right|}" loading="lazy">, which is related to the Ising Hamiltonian as follows,

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=\sum _{ij\in E(G)}W_{ij}-4\left|\delta (V^{+})\right|.}</annotation>

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

A significant number of statistical questions to ask about this model are in the limit of large numbers of spins:

<ul><li>In a typical configuration, are most of the spins +1 or −1, or are they split equally?</li>

<li>If a spin at any given position i is 1, what is the probability that the spin at position j is also 1?</li>

<li>If β is changed, is there a phase transition?</li>

<li>On a lattice Λ, what is the fractal dimension of the shape of a large cluster of +1 spins?</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Basic_properties_and_history">Basic properties and history</h2></div>

The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = Zd, Jij = 1, h = 0.

<div class="mw-heading mw-heading3"><h3 id="No_phase_transition_in_one_dimension">No phase transition in one dimension</h3></div>

In his 1924 PhD thesis, Ising solved the model for the d = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no <a href="Phase_transition" title="Phase transition">phase transition</a>.<a href="#cite_note-6">[6]</a> Namely, for any positive β, the correlations ⟨σiσj⟩ decay exponentially in |i − j|:

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{i}\sigma _{j}\rangle _{\beta }\leq C\exp \left(-c(\beta )|i-j|\right),}</annotation>

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and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.

<div class="mw-heading mw-heading3"><h3 id="Phase_transition_and_exact_solution_in_two_dimensions">Phase transition and exact solution in two dimensions</h3></div>

The Ising model undergoes a <a href="Phase_transition" title="Phase transition">phase transition</a> between an <a href="Ordered_phase" class="mw-redirect" title="Ordered phase">ordered</a> and a <a href="Disordered_phase" class="mw-redirect" title="Disordered phase">disordered phase</a> in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{i}\sigma _{j}\rangle _{\beta }\geq c(\beta )>0.}</annotation>

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This was first proven by <a href="Rudolf_Peierls" title="Rudolf Peierls">Rudolf Peierls</a> in 1936,<a href="#cite_note-7">[7]</a> using what is now called a Peierls argument.

The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by <a href="Lars_Onsager" title="Lars Onsager">Lars Onsager</a> (<a href="#CITEREFOnsager1944">1944</a>). Onsager obtained the <a href="Correlation_function" title="Correlation function">correlation functions</a> and <a href="Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> of the Ising model and announced the formula for the <a href="Spontaneous_magnetization" title="Spontaneous magnetization">spontaneous magnetization</a> for the 2-dimensional model in 1949 but did not give a derivation. <a href="#CITEREFYang1952">Yang (1952)</a> gave the first published proof of this formula, using a <a href="Szeg%C5%91_limit_theorems" title="Szegő limit theorems">limit formula</a> for <a href="Fredholm_determinant" title="Fredholm determinant">Fredholm determinants</a>, proved in 1951 by <a href="G%C3%A1bor_Szeg%C5%91" title="Gábor Szegő">Szegő</a> in direct response to Onsager's work.<a href="#cite_note-Montroll_1963_pages=308-309-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Correlation_inequalities">Correlation inequalities</h3></div>

A number of <a href="Correlation_inequality" title="Correlation inequality">correlation inequalities</a> have been derived rigorously for the Ising spin correlations (for general lattice structures), which have enabled mathematicians to study the Ising model both on and off criticality.

<div class="mw-heading mw-heading4"><h4 id="Griffiths_inequality">Griffiths inequality</h4></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Griffiths_inequality" title="Griffiths inequality">Griffiths inequality</a></div>

Given any subset of spins <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{A}}">

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

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</math><img src="./399e9d09fb7fc7fc1f286c97d3bde5ef6a814fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.792ex; height:2.009ex;" alt="{\displaystyle \sigma _{A}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{B}}">

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

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</math><img src="./6daf3fafcc7df4a3fda4f23f5c09b9c3944c346f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.807ex; height:2.009ex;" alt="{\displaystyle \sigma _{B}}" loading="lazy"> on the lattice, the following inequality holds,

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{A}\sigma _{B}\rangle \geq \langle \sigma _{A}\rangle \langle \sigma _{B}\rangle ,}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \sigma _{A}\rangle =\langle \prod _{j\in A}\sigma _{j}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{A}\rangle =\langle \prod _{j\in A}\sigma _{j}\rangle }</annotation>

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</math><img src="./f7254bf60d457a83ce6801e47d544952544686c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.14ex; height:6.009ex;" alt="{\displaystyle \langle \sigma _{A}\rangle =\langle \prod _{j\in A}\sigma _{j}\rangle }" loading="lazy">.

With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\emptyset }">

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

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</math><img src="./a3bac88b39d1d65791833cc8bf9659a01a44430e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.025ex; height:2.509ex;" alt="{\displaystyle B=\emptyset }" loading="lazy">, the special case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \sigma _{A}\rangle \geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{A}\rangle \geq 0}</annotation>

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This means that spins are positively correlated on the Ising ferromagnet. An immediate application of this is that the magnetization of any set of spins <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \sigma _{A}\rangle }">

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</math><img src="./144b6bb8bcf0d5b0ab8da4eec76febee752ee286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.601ex; height:2.843ex;" alt="{\displaystyle \langle \sigma _{A}\rangle }" loading="lazy"> is increasing with respect to any set of coupling constants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{B}}">

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

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</math><img src="./b58f5463d5483d92900f34f76dad9a1d27f4454b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.77ex; height:2.509ex;" alt="{\displaystyle J_{B}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Simon-Lieb_inequality">Simon-Lieb inequality</h4></div>

The Simon-Lieb inequality<a href="#cite_note-9">[9]</a> states that for any set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> disconnecting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> (e.g. the boundary of a box with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> being inside the box and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{x}\sigma _{y}\rangle \leq \sum _{z\in S}\langle \sigma _{x}\sigma _{z}\rangle \langle \sigma _{z}\sigma _{y}\rangle .}</annotation>

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This inequality can be used to establish the sharpness of phase transition for the Ising model.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading4"><h4 id="FKG_inequality">FKG inequality</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="FKG_inequality" title="FKG inequality">FKG inequality</a></div>

This inequality is proven first for a type of <a href="Random_cluster_model" title="Random cluster model">positively-correlated percolation model</a>, of which includes a representation of the Ising model. It is used to determine the critical temperatures of planar <a href="Potts_model" title="Potts model">Potts model</a> using percolation arguments (which includes the Ising model as a special case).<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Historical_significance">Historical significance</h2></div>

While the laws of chemical bonding made it clear to nineteenth century chemists that atoms were real, among physicists the debate continued well into the early twentieth century. Atomists, notably <a href="James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> and <a href="Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>, applied Hamilton's formulation of Newton's laws to large systems, and found that the <a href="Statistical_mechanics" title="Statistical mechanics">statistical behavior</a> of the atoms correctly describes room temperature gases. But classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gases at low temperature.

Once modern <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> was formulated, atomism was no longer in conflict with experiment, but this did not lead to a universal acceptance of statistical mechanics, which went beyond atomism. <a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a> had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics. But many faulty arguments survived from the 19th century, when statistical mechanics was considered dubious. The lapses in intuition mostly stemmed from the fact that the limit of an infinite statistical system has many <a href="Zero%E2%80%93one_law" title="Zero–one law">zero-one laws</a> which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the overall, aggregate behavior.

<div class="mw-heading mw-heading3"><h3 id="No_phase_transitions_in_finite_volume">No phase transitions in finite volume</h3></div>

In the early part of the twentieth century, some believed that the <a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> could never describe a phase transition, based on the following argument:

<ol><li>The partition function is a sum of e−βE over all configurations.</li>

<li>The exponential function is everywhere <a href="Analytic_function" title="Analytic function">analytic</a> as a function of β.</li>

<li>The sum of analytic functions is an analytic function.</li></ol>

This argument works for a finite sum of exponentials, and correctly establishes that there are no singularities in the free energy of a system of a finite size. For systems which are in the thermodynamic limit (that is, for infinite systems) the infinite sum can lead to singularities. The convergence to the thermodynamic limit is fast, so that the phase behavior is apparent already on a relatively small lattice, even though the singularities are smoothed out by the system's finite size.

This was first established by <a href="Rudolf_Peierls" title="Rudolf Peierls">Rudolf Peierls</a> in the Ising model.

<div class="mw-heading mw-heading3"><h3 id="Peierls_droplets">Peierls droplets</h3></div>

Shortly after Lenz and Ising constructed the Ising model, Peierls was able to explicitly show that a phase transition occurs in two dimensions.

To do this, he compared the high-temperature and low-temperature limits. At infinite temperature (β = 0) all configurations have equal probability. Each spin is completely independent of any other, and if typical configurations at infinite temperature are plotted so that plus/minus are represented by black and white, they look like <a href="Noise_(video)" title="Noise (video)">television snow</a>. For high, but not infinite temperature, there are small correlations between neighboring positions, the snow tends to clump a little bit, but the screen stays randomly looking, and there is no net excess of black or white.

A quantitative measure of the excess is the magnetization, which is the average value of the spin:

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<annotation encoding="application/x-tex">{\displaystyle M={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}.}</annotation>

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A bogus argument analogous to the argument in the last section now establishes that the average magnetization in the Ising model is always zero.

<ol><li>Every configuration of spins has equal energy to the configuration with all spins flipped.</li>

<li>So for every configuration with magnetization M there is a configuration with magnetization −M with equal probability.</li>

<li>The system should therefore spend equal amounts of time in the configuration with magnetization M as with magnetization −M.</li>

<li>So the average magnetization (over all time) is zero.</li></ol>

As before, this only proves that the average magnetization is zero at any finite volume. For an infinite system, fluctuations might not be able to push the system from a mostly plus state to a mostly minus with a nonzero probability.

For very high temperatures, the magnetization is zero, as it is at infinite temperature. To see this, note that if spin A has only a small correlation ε with spin B, and B is only weakly correlated with C, but C is otherwise independent of A, the amount of correlation of A and C goes like ε2. For two spins separated by distance L, the amount of correlation goes as εL, but if there is more than one path by which the correlations can travel, this amount is enhanced by the number of paths.

The number of paths of length L on a square lattice in d dimensions is

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since there are 2d choices for where to go at each step.

A bound on the total correlation is given by the contribution to the correlation by summing over all paths linking two points, which is bounded above by the sum over all paths of length L divided by

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which goes to zero when ε is small.

At low temperatures (β ≫ 1) the configurations are near the lowest-energy configuration, the one where all the spins are plus or all the spins are minus. Peierls asked whether it is statistically possible at low temperature, starting with all the spins minus, to fluctuate to a state where most of the spins are plus. For this to happen, droplets of plus spin must be able to congeal to make the plus state.

The energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet L, where plus spins and minus spins neighbor each other. For a droplet with perimeter L, the area is somewhere between (L − 2)/2 (the straight line) and (L/4)2 (the square box). The probability cost for introducing a droplet has the factor e−βL, but this contributes to the partition function multiplied by the total number of droplets with perimeter L, which is less than the total number of paths of length L:

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<annotation encoding="application/x-tex">{\displaystyle N(L)<4^{2L}.}</annotation>

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So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by

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<annotation encoding="application/x-tex">{\displaystyle \sum _{L}L^{2}4^{2L}e^{-4\beta L},}</annotation>

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</math>

which goes to zero at large β. For β sufficiently large, this exponentially suppresses long loops, so that they cannot occur, and the magnetization never fluctuates too far from −1.

So Peierls established that the magnetization in the Ising model eventually defines <a href="Superselection_sector" class="mw-redirect" title="Superselection sector">superselection sectors</a>, separated domains not linked by finite fluctuations.

<div class="mw-heading mw-heading3"><h3 id="Kramers–Wannier_duality">Kramers–Wannier duality</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Kramers%E2%80%93Wannier_duality" title="Kramers–Wannier duality">Kramers–Wannier duality</a></div>

Kramers and Wannier were able to show that the high-temperature expansion and the low-temperature expansion of the model are equal up to an overall rescaling of the free energy. This allowed the phase-transition point in the two-dimensional model to be determined exactly (under the assumption that there is a unique critical point).

<div class="mw-heading mw-heading3"><h3 id="Yang–Lee_zeros">Yang–Lee zeros</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Lee%E2%80%93Yang_theorem" title="Lee–Yang theorem">Lee–Yang theorem</a></div>

After Onsager's solution, Yang and Lee investigated the way in which the partition function becomes singular as the temperature approaches the critical temperature.

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

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

The original motivation for the model was the phenomenon of <a href="Ferromagnetism" title="Ferromagnetism">ferromagnetism</a>. Iron is magnetic; once it is magnetized it stays magnetized for a long time compared to any atomic time.

In the 19th century, it was thought that magnetic fields are due to currents in matter, and <a href="Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a> postulated that permanent magnets are caused by permanent atomic currents. The motion of classical charged particles could not explain permanent currents though, as shown by <a href="Joseph_Larmor" title="Joseph Larmor">Larmor</a>. In order to have ferromagnetism, the atoms must have permanent <a href="Magnetic_moment" title="Magnetic moment">magnetic moments</a> which are not due to the motion of classical charges.

Once the electron's spin was discovered, it was clear that the magnetism should be due to a large number of electron spins all oriented in the same direction. It was natural to ask how the electrons' spins all know which direction to point in, because the electrons on one side of a magnet don't directly interact with the electrons on the other side. They can only influence their neighbors. The Ising model was designed to investigate whether a large fraction of the electron spins could be oriented in the same direction using only local forces.

<div class="mw-heading mw-heading3"><h3 id="Lattice_gas">Lattice gas</h3></div>

The Ising model can be reinterpreted as a statistical model for the motion of atoms. Since the kinetic energy depends only on momentum and not on position, while the statistics of the positions only depends on the potential energy, the thermodynamics of the gas only depends on the potential energy for each configuration of atoms.

A coarse model is to make space-time a lattice and imagine that each position either contains an atom or it doesn't. The space of configuration is that of independent bits Bi, where each bit is either 0 or 1 depending on whether the position is occupied or not. An attractive interaction reduces the energy of two nearby atoms. If the attraction is only between nearest neighbors, the energy is reduced by −4JBiBj for each occupied neighboring pair.

The density of the atoms can be controlled by adding a <a href="Chemical_potential" title="Chemical potential">chemical potential</a>, which is a multiplicative probability cost for adding one more atom. A multiplicative factor in probability can be reinterpreted as an additive term in the logarithm – the energy. The extra energy of a configuration with N atoms is changed by μN. The probability cost of one more atom is a factor of exp(−βμ).

So the energy of the lattice gas is:

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<annotation encoding="application/x-tex">{\displaystyle E=-{\frac {1}{2}}\sum _{\langle i,j\rangle }4JB_{i}B_{j}+\sum _{i}\mu B_{i}.}</annotation>

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</math>

Rewriting the bits in terms of spins, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}=(S_{i}+1)/2.}">

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<annotation encoding="application/x-tex">{\displaystyle B_{i}=(S_{i}+1)/2.}</annotation>

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</math><img src="./416338fa051f525273917172b534d1d52bc1212f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.671ex; height:2.843ex;" alt="{\displaystyle B_{i}=(S_{i}+1)/2.}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle E=-{\frac {1}{2}}\sum _{\langle i,j\rangle }JS_{i}S_{j}-{\frac {1}{2}}\sum _{i}(4J-\mu )S_{i}.}</annotation>

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</math>

For lattices where every site has an equal number of neighbors, this is the Ising model with a magnetic field h = (zJ − μ)/2, where z is the number of neighbors.

In biological systems, modified versions of the lattice gas model have been used to understand a range of binding behaviors. These include the binding of ligands to receptors in the cell surface,<a href="#cite_note-12">[12]</a> the binding of <a href="Chemotaxis" title="Chemotaxis">chemotaxis</a> proteins to the flagellar motor,<a href="#cite_note-13">[13]</a> and the condensation of DNA.<a href="#cite_note-14">[14]</a>

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

The activity of <a href="Neuron" title="Neuron">neurons</a> in the brain can be modelled statistically. Each neuron at any time is either active + or inactive −. The active neurons are those that send an <a href="Action_potential" title="Action potential">action potential</a> down the axon in any given time window, and the inactive ones are those that do not.

Following the general approach of Jaynes,<a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a> a later interpretation of Schneidman, Berry, Segev and Bialek,<a href="#cite_note-17">[17]</a>

is that the Ising model is useful for any model of neural function, because a statistical model for neural activity should be chosen using the <a href="Principle_of_maximum_entropy" title="Principle of maximum entropy">principle of maximum entropy</a>. Given a collection of neurons, a statistical model which can reproduce the average firing rate for each neuron introduces a <a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a> for each neuron:

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<annotation encoding="application/x-tex">{\displaystyle E=-\sum _{i}h_{i}S_{i}}</annotation>

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But the activity of each neuron in this model is statistically independent. To allow for pair correlations, when one neuron tends to fire (or not to fire) along with another, introduce pair-wise lagrange multipliers:

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<annotation encoding="application/x-tex">{\displaystyle E=-{\tfrac {1}{2}}\sum _{ij}J_{ij}S_{i}S_{j}-\sum _{i}h_{i}S_{i}}</annotation>

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}}">

<msub>

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

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</math><img src="./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"> are not restricted to neighbors. Note that this generalization of Ising model is sometimes called the quadratic exponential binary distribution in statistics.

This energy function only introduces probability biases for a spin having a value and for a pair of spins having the same value. Higher order correlations are unconstrained by the multipliers. An activity pattern sampled from this distribution requires the largest number of bits to store in a computer, in the most efficient coding scheme imaginable, as compared with any other distribution with the same average activity and pairwise correlations. This means that Ising models are relevant to any system which is described by bits which are as random as possible, with constraints on the pairwise correlations and the average number of 1s, which frequently occurs in both the physical and social sciences.

<div class="mw-heading mw-heading3"><h3 id="Spin_glasses">Spin glasses</h3></div>

With the Ising model the so-called <a href="Spin_glasses" class="mw-redirect" title="Spin glasses">spin glasses</a> can also be described, by the usual Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H=-{\frac {1}{2}}\,\sum J_{i,k}\,S_{i}\,S_{k},}">

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<annotation encoding="application/x-tex">{\textstyle H=-{\frac {1}{2}}\,\sum J_{i,k}\,S_{i}\,S_{k},}</annotation>

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</math><img src="./8f0b2ad050d31106f718cc0abb7fb2e456fb0e29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.806ex; height:3.509ex;" alt="{\textstyle H=-{\frac {1}{2}}\,\sum J_{i,k}\,S_{i}\,S_{k},}" loading="lazy"> where the S-variables describe the Ising spins, while the Ji,k are taken from a random distribution. For spin glasses a typical distribution chooses antiferromagnetic bonds with probability p and ferromagnetic bonds with probability 1 − p (also known as the random-bond Ising model). These bonds stay fixed or "quenched" even in the presence of thermal fluctuations. When p = 0 we have the original Ising model. This system deserves interest in its own; particularly one has "non-ergodic" properties leading to strange relaxation behaviour. Much attention has been also attracted by the related bond and site dilute Ising model, especially in two dimensions, leading to intriguing critical behavior.<a href="#cite_note-18">[18]</a>

<div class="mw-heading mw-heading3"><h3 id="Artificial_neural_network">Artificial neural network</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Hopfield_network" title="Hopfield network">Hopfield network</a></div>

Ising model was instrumental in the development of the <a href="Hopfield_network" title="Hopfield network">Hopfield network</a>. The original Ising model is a model for equilibrium. <a href="Roy_J._Glauber" title="Roy J. Glauber">Roy J. Glauber</a> in 1963 studied the Ising model evolving in time, as a process towards thermal equilibrium (<a href="Glauber_dynamics" title="Glauber dynamics">Glauber dynamics</a>), adding in the component of time.<a href="#cite_note-:222-19">[19]</a> (Kaoru Nakano, 1971)<a href="#cite_note-Nakano1971-20">[20]</a><a href="#cite_note-Nakano1972-21">[21]</a> and (<a href="Shun'ichi_Amari" title="Shun'ichi Amari">Shun'ichi Amari</a>, 1972),<a href="#cite_note-Amari19722-22">[22]</a> proposed to modify the weights of an Ising model by <a href="Hebbian_theory" title="Hebbian theory">Hebbian learning</a> rule as a model of associative memory. The same idea was published by (William A. Little, 1974),<a href="#cite_note-little74-23">[23]</a> who was cited by Hopfield in his 1982 paper.

The <a href="Spin_glass#Sherrington–Kirkpatrick_model" title="Spin glass">Sherrington–Kirkpatrick model</a> of spin glass, published in 1975,<a href="#cite_note-24">[24]</a> is the Hopfield network with random initialization. Sherrington and Kirkpatrick found that it is highly likely for the energy function of the SK model to have many local minima. In the 1982 paper, Hopfield applied this recently developed theory to study the Hopfield network with binary activation functions.<a href="#cite_note-Hopfield1982-25">[25]</a> In a 1984 paper he extended this to continuous activation functions.<a href="#cite_note-:03-26">[26]</a> It became a standard model for the study of neural networks through statistical mechanics.<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a>

The <a href="Melt_pond" title="Melt pond">melt pond</a> can be modelled by the Ising model; sea ice topography data bears rather heavily on the results. The state variable is binary for a simple 2D approximation, being either water or ice.<a href="#cite_note-29">[29]</a>

<div class="mw-heading mw-heading3"><h3 id="Cayley_tree_topologies_and_large_neural_networks">Cayley tree topologies and large neural networks</h3></div>

In order to investigate an Ising model with potential relevance for large (e.g. with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{4}}">

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<annotation encoding="application/x-tex">{\displaystyle 10^{4}}</annotation>

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</math><img src="./5033e1e7d25542f462483cf58441681847e04153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{4}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{5}}">

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 10^{5}}</annotation>

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</math><img src="./d7310d2ac4ddefd259b8a2c7c8fb89c6a9ea11a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{5}}" loading="lazy"> interactions per node) neural nets, at the suggestion of Krizan in 1979, <a href="#CITEREFBarth1981">Barth (1981)</a> obtained the exact analytical expression for the free energy of the Ising model on the closed <a href="Cayley_tree" class="mw-redirect" title="Cayley tree">Cayley tree</a> (with an arbitrarily large branching ratio) for a zero-external magnetic field (in the thermodynamic limit) by applying the methodologies of <a href="#CITEREFGlasser1970">Glasser (1970)</a> and <a href="#CITEREFJellito1979">Jellito (1979)</a>

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<annotation encoding="application/x-tex">{\displaystyle -\beta f=\ln 2+{\frac {2\gamma }{(\gamma +1)}}\ln(\cosh J)+{\frac {\gamma (\gamma -1)}{(\gamma +1)}}\sum _{i=2}^{z}{\frac {1}{\gamma ^{i}}}\ln J_{i}(\tau )}</annotation>

</semantics>

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

</mstyle>

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

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is an arbitrary branching ratio (greater than or equal to 2), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\equiv \tanh J}">

</mstyle>

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

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</math><img src="./b4df9f6c59d28249333310140eb6f011bb65a909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.449ex; height:2.176ex;" alt="{\displaystyle t\equiv \tanh J}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \equiv t^{2}}">

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

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</math><img src="./0967ba35f12a324f6425d5381b3255e13d21c14e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.194ex; height:2.676ex;" alt="{\displaystyle \tau \equiv t^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\equiv \beta \epsilon }">

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

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</math><img src="./63bd3b955ffa30e9243f11b2dd8e1e1741e0bde0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.846ex; height:2.509ex;" alt="{\displaystyle J\equiv \beta \epsilon }" loading="lazy"> (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

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

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</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"> representing the nearest-neighbor interaction energy) and there are k (→ ∞ in the thermodynamic limit) generations in each of the tree branches (forming the closed tree architecture as shown in the given closed Cayley tree diagram.) The sum in the last term can be shown to converge uniformly and rapidly (i.e. for z → ∞, it remains finite) yielding a continuous and monotonous function, establishing that, for <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="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> greater than or equal to 2, the free energy is a continuous function of temperature T. Further analysis of the free energy indicates that it exhibits an unusual discontinuous first derivative at the critical temperature (<a href="#CITEREFKrizanBarthGlasser1983">Krizan, Barth & Glasser (1983)</a>, <a href="#CITEREFGlasserGoldberg1983">Glasser & Goldberg (1983)</a>.)

The spin-spin correlation between sites (in general, m and n) on the tree was found to have a transition point when considered at the vertices (e.g. A and Ā, its reflection), their respective neighboring sites (such as B and its reflection), and between sites adjacent to the top and bottom extreme vertices of the two trees (e.g. A and B), as may be determined from

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<annotation encoding="application/x-tex">{\displaystyle \langle s_{m}s_{n}\rangle ={Z_{N}}^{-1}(0,T)[\cosh J]^{N_{b}}2^{N}\sum _{l=1}^{z}g_{mn}(l)t^{l}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{b}}">

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</math><img src="./72f4f985ecf182d4503ee6fba5942e91514b4d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle N_{b}}" loading="lazy"> is equal to the number of bonds, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{mn}(l)t^{l}}">

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</math><img src="./4893010a4342e759a248b844176ec482a663b0ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.835ex; height:3.176ex;" alt="{\displaystyle g_{mn}(l)t^{l}}" loading="lazy"> is the number of graphs counted for odd vertices with even intermediate sites (see cited methodologies and references for detailed calculations), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{N}}">

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</math><img src="./f3cc22b5fa0e34487c8a6153965408e004c6e253.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.854ex; height:2.676ex;" alt="{\displaystyle 2^{N}}" loading="lazy"> is the multiplicity resulting from two-valued spin possibilities and the partition function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {Z_{N}}}">

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</math><img src="./78c5cad588ccc21113fbabee2f957c8657dd6ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.279ex; height:2.509ex;" alt="{\displaystyle {Z_{N}}}" loading="lazy"> is derived from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\{s\}}e^{-\beta H}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{\{s\}}e^{-\beta H}}</annotation>

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</math><img src="./2e7a1bbf171f8eb54a7220d333eda5f84986679b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:8.737ex; height:6.009ex;" alt="{\displaystyle \sum _{\{s\}}e^{-\beta H}}" loading="lazy">. (Note: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cfda82668232cbdc0874ed28ab8b6079420d1ffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}" loading="lazy"> is consistent with the referenced literature in this section and is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./de6e810a93f67802ecb603ee0e3324005c6e583e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6ab3208a7d0c634ef720e03ff5a9949e8310edc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}" loading="lazy"> utilized above and in earlier sections; it is valued at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0bfeaa85da53ad1947d8000926cfea33827ef1e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}" loading="lazy">.) The critical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{C}}">

<msub>

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</msub>

</mstyle>

</mrow>

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

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</math><img src="./becff78cd94880c5e7354a84d31b69dae139a7d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.839ex; height:2.509ex;" alt="{\displaystyle T_{C}}" loading="lazy"> is given by

<msub>

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<mfrac>

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</mrow>

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<msqrt>

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<msqrt>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle T_{C}={\frac {2\epsilon }{k_{\text{B}}[\ln({\sqrt {\gamma }}+1)-\ln({\sqrt {\gamma }}-1)]}}.}</annotation>

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</math>

The critical temperature for this model is only determined by the branching ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> and the site-to-site interaction energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy">, a fact which may have direct implications associated with neural structure vs. its function (in that it relates the energies of interaction and branching ratio to its transitional behavior.) For example, a relationship between the transition behavior of activities of neural networks between sleeping and wakeful states (which may correlate with a spin-spin type of phase transition) in terms of changes in neural interconnectivity (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy">) and/or neighbor-to-neighbor interactions (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy">), over time, is just one possible avenue suggested for further experimental investigation into such a phenomenon. In any case, for this Ising model it was established, that “the stability of the long-range correlation increases with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> or increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy">.”

For this topology, the spin-spin correlation was found to be zero between the extreme vertices and the central sites at which the two trees (or branches) are joined (i.e. between A and individually C, D, or E.) This behavior is explained to be due to the fact that, as k increases, the number of links increases exponentially (between the extreme vertices) and so even though the contribution to spin correlations decrease exponentially, the correlation between sites such as the extreme vertex (A) in one tree and the extreme vertex in the joined tree (Ā) remains finite (above the critical temperature.) In addition, A and B also exhibit a non-vanishing correlation (as do their reflections) thus lending itself to, for B level sites (with A level), being considered “clusters” which tend to exhibit synchronization of firing.

Based upon a review of other classical network models as a comparison, the Ising model on a closed Cayley tree was determined to be the first classical statistical mechanical model to demonstrate both local and long-range sites with non-vanishing spin-spin correlations, while at the same time exhibiting intermediate sites with zero correlation, which indeed was a relevant matter for large neural networks at the time of its consideration. The model's behavior is also of relevance for any other divergent-convergent tree physical (or biological) system exhibiting a closed Cayley tree topology with an Ising-type of interaction. This topology should not be ignored since its behavior for Ising models has been solved exactly, and presumably nature will have found a way of taking advantage of such simple symmetries at many levels of its designs.

<a href="#CITEREFBarth1981">Barth (1981)</a> early on noted the possibility of interrelationships between (1) the classical large neural network model (with similar coupled divergent-convergent topologies) with (2) an underlying statistical quantum mechanical model (independent of topology and with persistence in fundamental quantum states):

/* start https://en.wikipedia.org/ */

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</style><blockquote class="templatequote">The most significant result obtained from the closed Cayley tree model involves the occurrence of long-range correlation in the absence of intermediate-range correlation. This result has not been demonstrated by other classical models. The failure of the classical view of impulse transmission to account for this phenomenon has been cited by numerous investigators (Ricciiardi and Umezawa, 1967, Hokkyo 1972, Stuart, Takahashi and Umezawa 1978, 1979) as significant enough to warrant radically new assumptions on a very fundamental level and have suggested the existence of quantum cooperative modes within the brain…In addition, it is interesting to note that the (modeling) of…Goldstone particles or bosons (as per Umezawa, et al)…within the brain, demonstrates the long-range correlation of quantum numbers preserved in the ground state…In the closed Cayley tree model ground states of pairs of sites, as well as the state variable of individual sites, (can) exhibit long-range correlation.</blockquote>

It was a natural and common belief among early neurophysicists (e.g. Umezawa, Krizan, Barth, etc.) that classical neural models (including those with statistical mechanical aspects) will one day have to be integrated with quantum physics (with quantum statistical aspects), similar perhaps to how the domain of chemistry has historically integrated itself into quantum physics via quantum chemistry.

Several additional statistical mechanical problems of interest remain to be solved for the closed Cayley tree, including the time-dependent case and the external field situation, as well as theoretical efforts aimed at understanding interrelationships with underlying quantum constituents and their physics.

<div class="mw-heading mw-heading2"><h2 id="Numerical_simulation">Numerical simulation</h2></div>

The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising model with

<dl><dd>L = |Λ|: the total number of sites on the lattice,</dd>

<dd>σj ∈ {−1, +1}: an individual spin site on the lattice, j = 1, ..., L,</dd>

<dd>S ∈ {−1, +1}L: state of the system.</dd></dl>

Since every spin site has ±1 spin, there are 2L different states that are possible.<a href="#cite_note-Newman-30">[30]</a> This motivates the reason for the Ising model to be simulated using <a href="Monte_Carlo_methods" class="mw-redirect" title="Monte Carlo methods">Monte Carlo methods</a>.<a href="#cite_note-Newman-30">[30]</a>

The <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> that is commonly used to represent the energy of the model when using Monte Carlo methods is:

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\sum _{\langle i~j\rangle }\sigma _{i}\sigma _{j}-h\sum _{j}\sigma _{j}.}</annotation>

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</math>

Furthermore, the Hamiltonian is further simplified by assuming zero external field h, since many questions that are posed to be solved using the model can be answered in absence of an external field. This leads us to the following energy equation for state σ:

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\sum _{\langle i~j\rangle }\sigma _{i}\sigma _{j}.}</annotation>

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</math>

Given this Hamiltonian, quantities of interest such as the specific heat or the magnetization of the magnet at a given temperature can be calculated.<a href="#cite_note-Newman-30">[30]</a>

<div class="mw-heading mw-heading3"><h3 id="Metropolis_algorithm">Metropolis algorithm</h3></div>

The <a href="Metropolis%E2%80%93Hastings_algorithm" title="Metropolis–Hastings algorithm">Metropolis–Hastings algorithm</a> is the most commonly used Monte Carlo algorithm to calculate Ising model estimations.<a href="#cite_note-Newman-30">[30]</a> The algorithm first chooses selection probabilities g(μ, ν), which represent the probability that state ν is selected by the algorithm out of all states, given that one is in state μ. It then uses acceptance probabilities A(μ, ν) so that <a href="Detailed_balance" title="Detailed balance">detailed balance</a> is satisfied. If the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it. If ν is not accepted then we stay in μ. This process is repeated until some stopping criterion is met, which for the Ising model is often when the lattice becomes <a href="Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a>, meaning all of the sites point in the same direction.<a href="#cite_note-Newman-30">[30]</a>

When implementing the algorithm, one must ensure that g(μ, ν) is selected such that <a href="Ergodicity" title="Ergodicity">ergodicity</a> is met. In <a href="Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a> a system's energy only fluctuates within a small range.<a href="#cite_note-Newman-30">[30]</a> This is the motivation behind the concept of single-spin-flip dynamics,<a href="#cite_note-pre0-31">[31]</a> which states that in each transition, we will only change one of the spin sites on the lattice.<a href="#cite_note-Newman-30">[30]</a> Furthermore, by using single- spin-flip dynamics, one can get from any state to any other state by flipping each site that differs between the two states one at a time. The maximum amount of change between the energy of the present state, Hμ and any possible new state's energy Hν (using single-spin-flip dynamics) is 2J between the spin we choose to "flip" to move to the new state and that spin's neighbor.<a href="#cite_note-Newman-30">[30]</a> Thus, in a 1D Ising model, where each site has two neighbors (left and right), the maximum difference in energy would be 4J. Let c represent the lattice coordination number; the number of nearest neighbors that any lattice site has. We assume that all sites have the same number of neighbors due to <a href="Periodic_boundary_conditions" title="Periodic boundary conditions">periodic boundary conditions</a>.<a href="#cite_note-Newman-30">[30]</a> It is important to note that the Metropolis–Hastings algorithm does not perform well around the critical point due to critical slowing down. Other techniques such as multigrid methods, Niedermayer's algorithm, <a href="Swendsen%E2%80%93Wang_algorithm" title="Swendsen–Wang algorithm">Swendsen–Wang algorithm</a>, or the <a href="Wolff_algorithm" title="Wolff algorithm">Wolff algorithm</a> are required in order to resolve the model near the critical point; a requirement for determining the critical exponents of the system.

Specifically for the Ising model and using single-spin-flip dynamics, one can establish the following. Since there are L total sites on the lattice, using single-spin-flip as the only way we transition to another state, we can see that there are a total of L new states ν from our present state μ. The algorithm assumes that the selection probabilities are equal to the L states: g(μ, ν) = 1/L. <a href="Detailed_balance" title="Detailed balance">Detailed balance</a> tells us that the following equation must hold:

<mfrac>

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<msup>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {P(\mu ,\nu )}{P(\nu ,\mu )}}={\frac {g(\mu ,\nu )A(\mu ,\nu )}{g(\nu ,\mu )A(\nu ,\mu )}}={\frac {A(\mu ,\nu )}{A(\nu ,\mu )}}={\frac {P_{\beta }(\nu )}{P_{\beta }(\mu )}}={\frac {{\frac {1}{Z}}e^{-\beta (H_{\nu })}}{{\frac {1}{Z}}e^{-\beta (H_{\mu })}}}=e^{-\beta (H_{\nu }-H_{\mu })}.}</annotation>

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Thus, we want to select the acceptance probability for our algorithm to satisfy

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A(\mu ,\nu )}{A(\nu ,\mu )}}=e^{-\beta (H_{\nu }-H_{\mu })}.}</annotation>

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</math>

If Hν > Hμ, then A(ν, μ) > A(μ, ν). Metropolis sets the larger of A(μ, ν) or A(ν, μ) to be 1. By this reasoning the acceptance algorithm is:<a href="#cite_note-Newman-30">[30]</a>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mu ,\nu )={\begin{cases}e^{-\beta (H_{\nu }-H_{\mu })},&{\text{if }}H_{\nu }-H_{\mu }>0,\\1&{\text{otherwise}}.\end{cases}}}">

<mrow>

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<mtext>otherwise</mtext>

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<annotation encoding="application/x-tex">{\displaystyle A(\mu ,\nu )={\begin{cases}e^{-\beta (H_{\nu }-H_{\mu })},&{\text{if }}H_{\nu }-H_{\mu }>0,\\1&{\text{otherwise}}.\end{cases}}}</annotation>

</semantics>

</math>

The basic form of the algorithm is as follows:

<ol><li>Pick a spin site using selection probability g(μ, ν) and calculate the contribution to the energy involving this spin.</li>

<li>Flip the value of the spin and calculate the new contribution.</li>

<li>If the new energy is less, keep the flipped value.</li>

<li>If the new energy is more, only keep with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\beta (H_{\nu }-H_{\mu })}.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{-\beta (H_{\nu }-H_{\mu })}.}</annotation>

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</math><img src="./c23ca132fe13b4a0a425d0cbf6618a2bd0a5e010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.313ex; height:2.843ex;" alt="{\displaystyle e^{-\beta (H_{\nu }-H_{\mu })}.}" loading="lazy"></li>

<li>Repeat.</li></ol>

The change in energy Hν − Hμ only depends on the value of the spin and its nearest graph neighbors. So if the graph is not too connected, the algorithm is fast. This process will eventually produce a pick from the distribution.

<div class="mw-heading mw-heading3"><h3 id="As_a_Markov_chain">As a Markov chain</h3></div>

It is possible to view the Ising model as a <a href="Markov_chain" title="Markov chain">Markov chain</a>, as the immediate probability Pβ(ν) of transitioning to a future state ν only depends on the present state μ. The Metropolis algorithm is actually a version of a <a href="Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> simulation, and since we use single-spin-flip dynamics in the Metropolis algorithm, every state can be viewed as having links to exactly L other states, where each transition corresponds to flipping a single spin site to the opposite value.<a href="#cite_note-32">[32]</a> Furthermore, since the energy equation Hσ change only depends on the nearest-neighbor interaction strength J, the Ising model and its variants such the <a href="Sznajd_model" title="Sznajd model">Sznajd model</a> can be seen as a form of a <a href="Contact_process_(mathematics)#Voter_model" title="Contact process (mathematics)">voter model</a> for opinion dynamics.

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

<div class="mw-heading mw-heading3"><h3 id="One_dimension">One dimension</h3></div>

The thermodynamic limit exists as long as the interaction decay is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}</annotation>

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</math><img src="./421012ac1e2e89f6212a35507a5b676ba336b0ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.323ex; height:3.343ex;" alt="{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}" loading="lazy"> with α > 1.<a href="#cite_note-Ruelle-33">[33]</a>

<ul><li>In the case of ferromagnetic interaction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}</annotation>

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</math><img src="./421012ac1e2e89f6212a35507a5b676ba336b0ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.323ex; height:3.343ex;" alt="{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}" loading="lazy"> with 1 < α < 2, Dyson proved, by comparison with the hierarchical case, that there is phase transition at small enough temperature.<a href="#cite_note-34">[34]</a></li>

<li>In the case of ferromagnetic interaction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}\sim |i-j|^{-2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle J_{ij}\sim |i-j|^{-2}}</annotation>

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</math><img src="./fc54e9ee7a1c7c352fa9a79ece4dae6f9806fca5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.093ex; height:3.509ex;" alt="{\displaystyle J_{ij}\sim |i-j|^{-2}}" loading="lazy">, Fröhlich and Spencer proved that there is phase transition at small enough temperature (in contrast with the hierarchical case).<a href="#cite_note-35">[35]</a></li>

<li>In the case of interaction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}</annotation>

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</math><img src="./421012ac1e2e89f6212a35507a5b676ba336b0ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.323ex; height:3.343ex;" alt="{\displaystyle J_{ij}\sim |i-j|^{-\alpha }}" loading="lazy"> with α > 2 (which includes the case of finite-range interactions), there is no phase transition at any positive temperature (i.e. finite β), since the <a href="Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> is analytic in the thermodynamic parameters.<a href="#cite_note-Ruelle-33">[33]</a></li>

<li>In the case of nearest neighbor interactions, E. Ising provided an exact solution of the model. At any positive temperature (i.e. finite β) the free energy is analytic in the thermodynamics parameters, and the truncated two-point spin correlation decays exponentially fast. At zero temperature (i.e. infinite β), there is a second-order phase transition: the free energy is infinite, and the truncated two-point spin correlation does not decay (remains constant). Therefore, T = 0 is the critical temperature of this case. Scaling formulas are satisfied.<a href="#cite_note-36">[36]</a></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Ising's_exact_solution">Ising's exact solution</h4></div>

In the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available. The Hamiltonian of the one-dimensional Ising model on a lattice of L sites with free boundary conditions is

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\sum _{i=1,\ldots ,L-1}\sigma _{i}\sigma _{i+1}-h\sum _{i}\sigma _{i},}</annotation>

</semantics>

</math>

where J and h can be any number, since in this simplified case J is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites. Then the

<a href="Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> is

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle f(\beta ,h)=-\lim _{L\to \infty }{\frac {1}{\beta L}}\ln Z(\beta )=-{\frac {1}{\beta }}\ln \left(e^{\beta J}\cosh \beta h+{\sqrt {e^{2\beta J}(\sinh \beta h)^{2}+e^{-2\beta J}}}\right),}</annotation>

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</math>

and the spin-spin correlation (i.e. the covariance) is

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{i}\sigma _{j}\rangle -\langle \sigma _{i}\rangle \langle \sigma _{j}\rangle =C(\beta )e^{-c(\beta )|i-j|},}</annotation>

</semantics>

</math>

where C(β) and c(β) are positive functions for T > 0. For T → 0, though, the inverse correlation length c(β) vanishes.

<div class="mw-heading mw-heading5"><h5 id="Proof">Proof</h5></div>

The proof of this result is a simple computation.

If h = 0, it is very easy to obtain the free energy in the case of free boundary condition, i.e. when

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\left(\sigma _{1}\sigma _{2}+\cdots +\sigma _{L-1}\sigma _{L}\right).}</annotation>

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</math>

Then the model factorizes under the change of variables

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<annotation encoding="application/x-tex">{\displaystyle \sigma '_{j}=\sigma _{j}\sigma _{j-1},\quad j\geq 2.}</annotation>

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</math>

This gives

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<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\sum _{\sigma _{1},\ldots ,\sigma _{L}}e^{\beta J\sigma _{1}\sigma _{2}}e^{\beta J\sigma _{2}\sigma _{3}}\cdots e^{\beta J\sigma _{L-1}\sigma _{L}}=2\prod _{j=2}^{L}\sum _{\sigma '_{j}}e^{\beta J\sigma '_{j}}=2\left[e^{\beta J}+e^{-\beta J}\right]^{L-1}.}</annotation>

</semantics>

</math>

Therefore, the free energy is

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle f(\beta ,0)=-{\frac {1}{\beta }}\ln \left[e^{\beta J}+e^{-\beta J}\right].}</annotation>

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</math>

With the same change of variables

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<annotation encoding="application/x-tex">{\displaystyle \langle \sigma _{j}\sigma _{j+N}\rangle =\left[{\frac {e^{\beta J}-e^{-\beta J}}{e^{\beta J}+e^{-\beta J}}}\right]^{N},}</annotation>

</semantics>

</math>

hence it decays exponentially as soon as T ≠ 0; but for T = 0, i.e. in the limit β → ∞ there is no decay.

If h ≠ 0 we need the transfer matrix method. For the periodic boundary conditions case is the following. The partition function is

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\sum _{\sigma _{1},\ldots ,\sigma _{L}}e^{\beta h\sigma _{1}}e^{\beta J\sigma _{1}\sigma _{2}}e^{\beta h\sigma _{2}}e^{\beta J\sigma _{2}\sigma _{3}}\cdots e^{\beta h\sigma _{L}}e^{\beta J\sigma _{L}\sigma _{1}}=\sum _{\sigma _{1},\ldots ,\sigma _{L}}V_{\sigma _{1},\sigma _{2}}V_{\sigma _{2},\sigma _{3}}\cdots V_{\sigma _{L},\sigma _{1}}.}</annotation>

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The coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\sigma ,\sigma '}}">

<msub>

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

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</math><img src="./d7534c2c3af7b6749d75f07c7e5f2936a6b2cc25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.458ex; height:2.843ex;" alt="{\displaystyle V_{\sigma ,\sigma '}}" loading="lazy"> can be seen as the entries of a matrix. There are different possible choices: a convenient one (because the matrix is symmetric) is

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{\sigma ,\sigma '}=e^{{\frac {\beta h}{2}}\sigma }e^{\beta J\sigma \sigma '}e^{{\frac {\beta h}{2}}\sigma '}}</annotation>

</semantics>

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or

<mrow>

<mtr>

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<annotation encoding="application/x-tex">{\displaystyle V={\begin{bmatrix}e^{\beta (h+J)}&e^{-\beta J}\\e^{-\beta J}&e^{-\beta (h-J)}\end{bmatrix}}.}</annotation>

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In matrix formalism

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\operatorname {Tr} \left(V^{L}\right)=\lambda _{1}^{L}+\lambda _{2}^{L}=\lambda _{1}^{L}\left[1+\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{L}\right],}</annotation>

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</math>

where λ1 is the highest eigenvalue of V, while λ2 is the other eigenvalue:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}=e^{\beta J}\cosh \beta h+{\sqrt {e^{2\beta J}(\cosh \beta h)^{2}-2\sinh 2\beta J}}=e^{\beta J}\cosh \beta h+{\sqrt {e^{2\beta J}(\sinh \beta h)^{2}+e^{-2\beta J}}},}</annotation>

</semantics>

</math>

and λ2 < λ1. This gives the formula of the free energy above. In the thermodynamics limit for the non-interaction case (J = 0), we got

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z_{N}\to (\lambda _{1})^{N}=(2\cosh \beta h)^{N},}</annotation>

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</math>

as the answer for the open-boundary Ising model.

<div class="mw-heading mw-heading5"><h5 id="Comments">Comments</h5></div>

The energy of the lowest state is −JL, when all the spins are the same. For any other configuration, the extra energy is equal to 2J times the number of sign changes that are encountered when scanning the configuration from left to right.

If we designate the number of sign changes in a configuration as k, the difference in energy from the lowest energy state is 2k. Since the energy is additive in the number of flips, the probability p of having a spin-flip at each position is independent. The ratio of the probability of finding a flip to the probability of not finding one is the Boltzmann factor:

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {p}{1-p}}=e^{-2\beta J}.}</annotation>

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</math>

The problem is reduced to independent biased <a href="Coin_toss" class="mw-redirect" title="Coin toss">coin tosses</a>. This essentially completes the mathematical description.

From the description in terms of independent tosses, the statistics of the model for long lines can be understood. The line splits into domains. Each domain is of average length exp(2β). The length of a domain is distributed exponentially, since there is a constant probability at any step of encountering a flip. The domains never become infinite, so a long system is never magnetized. Each step reduces the correlation between a spin and its neighbor by an amount proportional to p, so the correlations fall off exponentially.

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \langle S_{i}S_{j}\rangle \propto e^{-p|i-j|}.}</annotation>

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</math>

The <a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> is the volume of configurations, each configuration weighted by its Boltzmann weight. Since each configuration is described by the sign-changes, the partition function factorizes:

<mtext>configs</mtext>

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<annotation encoding="application/x-tex">{\displaystyle Z=\sum _{\text{configs}}e^{\sum _{k}S_{k}}=\prod _{k}(1+p)=(1+p)^{L}.}</annotation>

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</math>

The logarithm divided by L is the free energy density:

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<annotation encoding="application/x-tex">{\displaystyle \beta f=\log(1+p)=\log \left(1+{\frac {e^{-2\beta J}}{1+e^{-2\beta J}}}\right),}</annotation>

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</math>

which is <a href="Analytic_function" title="Analytic function">analytic</a> away from β = ∞. A sign of a <a href="Phase_transition" title="Phase transition">phase transition</a> is a non-analytic free energy, so the one-dimensional model does not have a phase transition.

<div class="mw-heading mw-heading4"><h4 id="One-dimensional_solution_with_transverse_field">One-dimensional solution with transverse field</h4></div>

To express the Ising Hamiltonian using a quantum mechanical description of spins, we replace the spin variables with their respective <a href="Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. However, depending on the direction of the magnetic field, we can create a transverse-field or longitudinal-field Hamiltonian. The <a href="Transverse-field_Ising_model" title="Transverse-field Ising model">transverse-field</a> Hamiltonian is given by

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-J\sum _{i=1,\ldots ,L}\sigma _{i}^{z}\sigma _{i+1}^{z}-h\sum _{i}\sigma _{i}^{x}.}</annotation>

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</math>

The transverse-field model experiences a phase transition between an ordered and disordered regime at J ~ h. This can be shown by a mapping of Pauli matrices

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{n}^{z}=\prod _{i=1}^{n}T_{i}^{x},}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{n}^{x}=T_{n}^{z}T_{n+1}^{z}.}</annotation>

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Upon rewriting the Hamiltonian in terms of this change-of-basis matrices, we obtain

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<annotation encoding="application/x-tex">{\displaystyle H(\sigma )=-h\sum _{i=1,\ldots ,L}T_{i}^{z}T_{i+1}^{z}-J\sum _{i}T_{i}^{x}.}</annotation>

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Since the roles of h and J are switched, the Hamiltonian undergoes a transition at J = h.<a href="#cite_note-Chakra-37">[37]</a>

<div class="mw-heading mw-heading4"><h4 id="Renormalization">Renormalization</h4></div>

When there is no external field, we can derive a functional equation that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\beta ,0)=f(\beta )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\beta ,0)=f(\beta )}</annotation>

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</math><img src="./18e3e4658f00240de8036d9144de9fbbaf104c6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.135ex; height:2.843ex;" alt="{\displaystyle f(\beta ,0)=f(\beta )}" loading="lazy"> satisfies using renormalization.<a href="#cite_note-38">[38]</a> Specifically, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{N}(\beta ,J)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z_{N}(\beta ,J)}</annotation>

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</math><img src="./4e66cd719561d3d90adaea8015230c4aacb47c79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.926ex; height:2.843ex;" alt="{\displaystyle Z_{N}(\beta ,J)}" loading="lazy"> be the partition function with <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="./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"> sites. Now we have:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{N}(\beta ,J)=\sum _{\sigma }e^{K\sigma _{2}(\sigma _{1}+\sigma _{3})}e^{K\sigma _{4}(\sigma _{3}+\sigma _{5})}\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle Z_{N}(\beta ,J)=\sum _{\sigma }e^{K\sigma _{2}(\sigma _{1}+\sigma _{3})}e^{K\sigma _{4}(\sigma _{3}+\sigma _{5})}\cdots }</annotation>

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</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K:=\beta J}">

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

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</math><img src="./06ed9e88f1088f981e81817a9593d2a1882c9a66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.615ex; height:2.509ex;" alt="{\displaystyle K:=\beta J}" loading="lazy">. We sum over each of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2},\sigma _{4},\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{2},\sigma _{4},\cdots }</annotation>

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</math><img src="./8c06ac77c1c334fc1e9d13d8236496452e924401.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.554ex; height:2.009ex;" alt="{\displaystyle \sigma _{2},\sigma _{4},\cdots }" loading="lazy">, to obtain<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{N}(\beta ,J)=\sum _{\sigma }(2\cosh(K(\sigma _{1}+\sigma _{3})))\cdot (2\cosh(K(\sigma _{3}+\sigma _{5})))\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle Z_{N}(\beta ,J)=\sum _{\sigma }(2\cosh(K(\sigma _{1}+\sigma _{3})))\cdot (2\cosh(K(\sigma _{3}+\sigma _{5})))\cdots }</annotation>

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</math>Now, since the cosh function is even, we can solve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ae^{K'\sigma _{1}\sigma _{3}}=2\cosh(K(\sigma _{1}+\sigma _{3}))}">

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<annotation encoding="application/x-tex">{\displaystyle Ae^{K'\sigma _{1}\sigma _{3}}=2\cosh(K(\sigma _{1}+\sigma _{3}))}</annotation>

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</math><img src="./4378b2b6231babe478e2255dd4571e4e27965729.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.951ex; height:3.343ex;" alt="{\displaystyle Ae^{K'\sigma _{1}\sigma _{3}}=2\cosh(K(\sigma _{1}+\sigma _{3}))}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A=2{\sqrt {\cosh(2K)}},K'={\frac {1}{2}}\ln \cosh(2K)}">

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<annotation encoding="application/x-tex">{\textstyle A=2{\sqrt {\cosh(2K)}},K'={\frac {1}{2}}\ln \cosh(2K)}</annotation>

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</math><img src="./6c93671d34ea8a3a5283ea5ee6a42caace4b1abf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.493ex; height:3.509ex;" alt="{\textstyle A=2{\sqrt {\cosh(2K)}},K'={\frac {1}{2}}\ln \cosh(2K)}" loading="lazy">. Now we have a self-similarity relation:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{N}}\ln Z_{N}(K)={\frac {1}{2}}\ln \left(2{\sqrt {\cosh(2K)}}\right)+{\frac {1}{2}}{\frac {1}{N/2}}\ln Z_{N/2}(K')}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{N}}\ln Z_{N}(K)={\frac {1}{2}}\ln \left(2{\sqrt {\cosh(2K)}}\right)+{\frac {1}{2}}{\frac {1}{N/2}}\ln Z_{N/2}(K')}</annotation>

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</math>Taking the limit, we obtain<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\beta )={\frac {1}{2}}\ln \left(2{\sqrt {\cosh(2K)}}\right)+{\frac {1}{2}}f(\beta ')}">

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<annotation encoding="application/x-tex">{\displaystyle f(\beta )={\frac {1}{2}}\ln \left(2{\sqrt {\cosh(2K)}}\right)+{\frac {1}{2}}f(\beta ')}</annotation>

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</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta 'J={\frac {1}{2}}\ln \cosh(2\beta J)}">

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<annotation encoding="application/x-tex">{\displaystyle \beta 'J={\frac {1}{2}}\ln \cosh(2\beta J)}</annotation>

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</math><img src="./42532a261eeb15d071f71d60e3284fbeeff4ebb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.482ex; height:5.176ex;" alt="{\displaystyle \beta 'J={\frac {1}{2}}\ln \cosh(2\beta J)}" loading="lazy">.

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

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

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is small, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\beta )\approx \ln 2}">

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<annotation encoding="application/x-tex">{\displaystyle f(\beta )\approx \ln 2}</annotation>

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</math><img src="./cb19500ba0ccbc4b5bd801bef761919d3dd57ed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.007ex; height:2.843ex;" alt="{\displaystyle f(\beta )\approx \ln 2}" loading="lazy">, so we can numerically evaluate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\beta )}">

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

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</math><img src="./6283fcad610b5a457f94e359b8cbd39d8b56d8e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.42ex; height:2.843ex;" alt="{\displaystyle f(\beta )}" loading="lazy"> by iterating the functional equation until <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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

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</math><img src="./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"> is small.

<div class="mw-heading mw-heading3"><h3 id="Two_dimensions">Two dimensions</h3></div>

In the ferromagnetic case there is a phase transition. At low temperature, the <a href="Peierls_argument" class="mw-redirect" title="Peierls argument">Peierls argument</a> proves positive magnetization for the nearest neighbor case and then, by the <a href="Griffiths_inequality" title="Griffiths inequality">Griffiths inequality</a>, also when longer range interactions are added. Meanwhile, at high temperature, the <a href="Cluster_expansion" title="Cluster expansion">cluster expansion</a> gives analyticity of the thermodynamic functions. In the nearest-neighbor case, the free energy was exactly computed by Onsager. The spin-spin correlation functions were computed by McCoy and Wu.

<div class="mw-heading mw-heading4"><h4 id="Onsager's_exact_solution">Onsager's exact solution</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Square_lattice_Ising_model" title="Square lattice Ising model">Square lattice Ising model</a></div>

<a href="#CITEREFOnsager1944">Onsager (1944)</a> obtained the following analytical expression for the free energy of the Ising model on the anisotropic square lattice when the magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=0}">

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

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</math><img src="./ffe239e1050529410001cc1c0b3245945bc69709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h=0}" loading="lazy"> in the thermodynamic limit as a function of temperature and the horizontal and vertical interaction energies <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><img src="./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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{2}}">

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

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</math><img src="./0f9986a8fbfd51097a5ff5e82d3252c9572b5835.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_{2}}" loading="lazy">, respectively

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<annotation encoding="application/x-tex">{\displaystyle -\beta f=\ln 2+{\frac {1}{8\pi ^{2}}}\int _{0}^{2\pi }d\theta _{1}\int _{0}^{2\pi }d\theta _{2}\ln[\cosh(2\beta J_{1})\cosh(2\beta J_{2})-\sinh(2\beta J_{1})\cos(\theta _{1})-\sinh(2\beta J_{2})\cos(\theta _{2})].}</annotation>

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</math>

From this expression for the free energy, all thermodynamic functions of the model can be calculated by using an appropriate derivative. The 2D Ising model was the first model to exhibit a continuous phase transition at a positive temperature. It occurs at the temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{c}}">

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

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</math><img src="./66261e3cbed8035b2bf7a9ccb878c786cf7556c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.302ex; height:2.509ex;" alt="{\displaystyle T_{c}}" loading="lazy"> which solves the equation

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<annotation encoding="application/x-tex">{\displaystyle \sinh \left({\frac {2J_{1}}{kT_{c}}}\right)\sinh \left({\frac {2J_{2}}{kT_{c}}}\right)=1.}</annotation>

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</math>

In the isotropic case when the horizontal and vertical interaction energies are equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{1}=J_{2}=J}">

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

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</math><img src="./bd4ea0906c10ccba84b561bb39e10a51f9ea90f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.357ex; height:2.509ex;" alt="{\displaystyle J_{1}=J_{2}=J}" loading="lazy">, the critical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{c}}">

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

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</math><img src="./66261e3cbed8035b2bf7a9ccb878c786cf7556c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.302ex; height:2.509ex;" alt="{\displaystyle T_{c}}" loading="lazy"> occurs at the following point

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<annotation encoding="application/x-tex">{\displaystyle T_{c}={\frac {2J}{k\ln(1+{\sqrt {2}})}}=(2.269185\cdots ){\frac {J}{k}}}</annotation>

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When the interaction energies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{1}}">

<msub>

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

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</math><img src="./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">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{2}}">

<msub>

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

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</math><img src="./0f9986a8fbfd51097a5ff5e82d3252c9572b5835.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_{2}}" loading="lazy"> are both negative, the Ising model becomes an antiferromagnet. Since the square lattice is bi-partite, it is invariant under this change when the magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=0}">

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

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</math><img src="./ffe239e1050529410001cc1c0b3245945bc69709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h=0}" loading="lazy">, so the free energy and critical temperature are the same for the antiferromagnetic case. For the triangular lattice, which is not bi-partite, the ferromagnetic and antiferromagnetic Ising model behave notably differently. Specifically, around a triangle, it is impossible to make all 3 spin-pairs antiparallel, so the antiferromagnetic Ising model cannot reach the minimal energy state. This is an example of <a href="Geometric_frustration" class="mw-redirect" title="Geometric frustration">geometric frustration</a>.

<div class="mw-heading mw-heading5"><h5 id="Onsager's_formula_for_spontaneous_magnetization">Onsager's formula for spontaneous magnetization</h5></div>

Onsager famously announced the following expression for the <a href="Spontaneous_magnetization" title="Spontaneous magnetization">spontaneous magnetization</a> M of a two-dimensional Ising ferromagnet on the square lattice at two different conferences in 1948, though without proof<a href="#cite_note-Montroll_1963_pages=308-309-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle M=\left(1-\left[\sinh 2\beta J_{1}\sinh 2\beta J_{2}\right]^{-2}\right)^{\frac {1}{8}}}</annotation>

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{1}}">

<msub>

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

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</math><img src="./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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{2}}">

<msub>

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

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</math><img src="./0f9986a8fbfd51097a5ff5e82d3252c9572b5835.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_{2}}" loading="lazy"> are horizontal and vertical interaction energies.

A complete derivation was only given in 1951 by <a href="#CITEREFYang1952">Yang (1952)</a> using a limiting process of transfer matrix eigenvalues. The proof was subsequently greatly simplified in 1963 by Montroll, Potts, and Ward<a href="#cite_note-Montroll_1963_pages=308-309-8">[8]</a> using <a href="G%C3%A1bor_Szeg%C5%91" title="Gábor Szegő">Szegő</a>'s <a href="Szeg%C5%91_limit_theorems" title="Szegő limit theorems">limit formula</a> for <a href="Toeplitz_determinant" class="mw-redirect" title="Toeplitz determinant">Toeplitz determinants</a> by treating the magnetization as the limit of correlation functions.

<div class="mw-heading mw-heading4"><h4 id="Minimal_model">Minimal model</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Two-dimensional_critical_Ising_model" title="Two-dimensional critical Ising model">Two-dimensional critical Ising model</a></div>

At the critical point, the two-dimensional Ising model is a <a href="Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">two-dimensional conformal field theory</a>. The spin and energy correlation functions are described by a <a href="Minimal_model_(physics)" title="Minimal model (physics)">minimal model</a>, which has been exactly solved.

<div class="mw-heading mw-heading3"><h3 id="Three_dimensions">Three dimensions</h3></div>

In three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field. Many theoreticians searched for an analytical three-dimensional solution for many decades, which would be analogous to Onsager's solution in the two-dimensional case.<a href="#cite_note-39">[39]</a> <a href="#cite_note-40">[40]</a> Such a solution has not been found until now, although there is no proof that it may not exist. In three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic strings by <a href="Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Alexander Polyakov</a> and Vladimir Dotsenko. This construction has been carried on the lattice, and the <a href="Continuum_limit" title="Continuum limit">continuum limit</a>, conjecturally describing the critical point, is unknown.

In three as in two dimensions, Peierls' argument shows that there is a phase transition. This phase transition is rigorously known to be continuous (in the sense that correlation length diverges and the magnetization goes to zero), and is called the <a href="Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical point</a>. It is believed that the critical point can be described by a renormalization group fixed point of the Wilson-Kadanoff renormalization group transformation. It is also believed that the phase transition can be described by a three-dimensional unitary conformal field theory, as evidenced by <a href="Metropolis%E2%80%93Hastings_algorithm" title="Metropolis–Hastings algorithm">Monte Carlo</a> simulations,<a href="#cite_note-41">[41]</a><a href="#cite_note-42">[42]</a> exact diagonalization results in quantum models,<a href="#cite_note-43">[43]</a> and quantum field theoretical arguments.<a href="#cite_note-44">[44]</a> Although it is an open problem to establish rigorously the renormalization group picture or the conformal field theory picture, theoretical physicists have used these two methods to compute the <a href="Critical_exponents" class="mw-redirect" title="Critical exponents">critical exponents</a> of the phase transition, which agree with the experiments and with the Monte Carlo simulations. This conformal field theory describing the three-dimensional Ising critical point is under active investigation using the method of the <a href="Conformal_bootstrap" title="Conformal bootstrap">conformal bootstrap</a>.<a href="#cite_note-45">[45]</a><a href="#cite_note-cmin-46">[46]</a><a href="#cite_note-SDPB-47">[47]</a><a href="#cite_note-Kadanoff-48">[48]</a> This method currently yields the most precise information about the structure of the critical theory (see <a href="Ising_critical_exponents" class="mw-redirect" title="Ising critical exponents">Ising critical exponents</a>).

In 2000, Sorin Istrail of <a href="Sandia_National_Laboratories" title="Sandia National Laboratories">Sandia National Laboratories</a> proved that the spin glass Ising model on a <a href="Nonplanar" class="mw-redirect" title="Nonplanar">nonplanar</a> lattice is <a href="NP-completeness" title="NP-completeness">NP-complete</a>. That is, assuming P ≠ NP, the general spin glass Ising model is exactly solvable only in <a href="Planar_graph" title="Planar graph">planar</a> cases, so solutions for dimensions higher than two are also intractable.<a href="#cite_note-49">[49]</a> Istrail's result only concerns the spin glass model with spatially varying couplings, and tells nothing about Ising's original ferromagnetic model with equal couplings.

<div class="mw-heading mw-heading3"><h3 id="Four_dimensions_and_above">Four dimensions and above</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="High-dimensional_Ising_model" title="High-dimensional Ising model">High-dimensional Ising model</a></div>

In any dimension, the Ising model can be productively described by a locally varying <a href="Mean_field_theory" class="mw-redirect" title="Mean field theory">mean field</a>. The field is defined as the average spin value over a large region, but not so large so as to include the entire system. The field still has slow variations from point to point, as the averaging volume moves. These fluctuations in the field are described by a continuum field theory in the infinite system limit. The accuracy of this approximation improves as the dimension becomes larger. A deeper understanding of how the Ising model behaves, going beyond mean-field approximations, can be achieved using <a href="Renormalization_group" title="Renormalization group">renormalization group</a> methods.

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<ul><li><a href="ANNNI_model" title="ANNNI model">ANNNI model</a></li>

<li><a href="Binder_parameter" title="Binder parameter">Binder parameter</a></li>

<li><a href="Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a></li>

<li><a href="Construction_of_an_irreducible_Markov_chain_in_the_Ising_model" title="Construction of an irreducible Markov chain in the Ising model">Construction of an irreducible Markov chain in the Ising model</a></li>

<li><a href="Geometrical_frustration" title="Geometrical frustration">Geometrical frustration</a></li>

<li><a href="Classical_Heisenberg_model" title="Classical Heisenberg model">Classical Heisenberg model</a></li>

<li><a href="Quantum_Heisenberg_model" title="Quantum Heisenberg model">Quantum Heisenberg model</a></li>

<li><a href="Kuramoto_model" title="Kuramoto model">Kuramoto model</a></li>

<li><a href="Maximal_evenness" title="Maximal evenness">Maximal evenness</a></li>

<li><a href="Order_operator" title="Order operator">Order operator</a></li>

<li><a href="Spin_model" title="Spin model">Spin model</a></li>

<li><a href="T-J_model" title="T-J model">t-J model</a></li>

<li><a href="Classical_XY_model" title="Classical XY model">Classical XY model</a></li>

<li><a href="ZN_model" title="ZN model">ZN model</a></li></ul>

</div>

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> See <a href="#CITEREFGallavotti1999">Gallavotti (1999)</a>, Chapters VI-VII.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a rel="nofollow" class="external text" href="http://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html">Ernst Ising, Contribution to the Theory of Ferromagnetism</a>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> See <a href="#CITEREFBaierlein1999">Baierlein (1999)</a>, Chapter 16.

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<li id="cite_note-SDPB-47"><a href="#cite_ref-SDPB_47-0">^</a> <cite id="CITEREFSimmons-Duffin2015" class="citation journal cs1">Simmons-Duffin, David (2015). "A semidefinite program solver for the conformal bootstrap". Journal of High Energy Physics. 2015 (6): 174. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1502.02033">1502.02033</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015JHEP...06..174S">2015JHEP...06..174S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP06%282015%29174">10.1007/JHEP06(2015)174</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1029-8479">1029-8479</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:35625559">35625559</a>.</cite>

</li>

<li id="cite_note-Kadanoff-48"><a href="#cite_ref-Kadanoff_48-0">^</a> <cite id="CITEREFKadanoff2014" class="citation journal cs1 cs1-prop-unfit">Kadanoff, Leo P. (April 30, 2014). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384">"Deep Understanding Achieved on the 3d Ising Model"</a>. Journal Club for Condensed Matter Physics. Archived from the original on July 22, 2015. Retrieved July 19, 2015.</cite>

</li>

<li id="cite_note-49"><a href="#cite_ref-49">^</a> <cite id="CITEREFCipra2000" class="citation journal cs1">Cipra, Barry A. (2000). <a rel="nofollow" class="external text" href="https://archive.siam.org/pdf/news/654.pdf">"The Ising Model Is NP-Complete"</a> (PDF). SIAM News. 33 (6).</cite>

</li>

</ol></div>

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

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<ul><li><cite id="CITEREFBarth1981" class="citation cs2"><a href="Peter_F._Barth" title="Peter F. Barth">Barth, P. F.</a> (1981), "Cooperativity and the Transition Behavior of Large Neural Nets", Master of Science Thesis, Burlington: University of Vermont: 1–118, <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/8231704">8231704</a></cite></li>

<li><cite id="CITEREFBaxter1982" class="citation cs2">Baxter, Rodney J. (1982), <a rel="nofollow" class="external text" href="https://physics.anu.edu.au/theophys/baxter_book.php">Exactly solved models in statistical mechanics</a>, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-083180-7</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0690578">0690578</a></cite></li>

<li><cite id="CITEREFK._Binder2001" class="citation cs2"><a href="Kurt_Binder" title="Kurt Binder">K. Binder</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Ising_model">"Ising model"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFBrush1967" class="citation journal cs1">Brush, Stephen G. (1967). "History of the Lenz-Ising Model". Reviews of Modern Physics. 39 (4): 883–893. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967RvMP...39..883B">1967RvMP...39..883B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.39.883">10.1103/RevModPhys.39.883</a>.</cite></li>

<li><cite id="CITEREFBaierlein1999" class="citation cs2">Baierlein, R. (1999), <a rel="nofollow" class="external text" href="https://archive.org/details/thermalphysics00ralp">Thermal Physics</a>, Cambridge: Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-59082-2</bdi></cite></li>

<li><cite id="CITEREFGallavotti1999" class="citation cs2"><a href="Giovanni_Gallavotti" title="Giovanni Gallavotti">Gallavotti, G.</a> (1999), Statistical mechanics, Texts and Monographs in Physics, Berlin: Springer-Verlag, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-03952-6">10.1007/978-3-662-03952-6</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-64883-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1707309">1707309</a></cite></li>

<li><cite id="CITEREFHuang1987" class="citation cs2"><a href="Kerson_Huang" title="Kerson Huang">Huang, Kerson</a> (1987), Statistical mechanics (2nd ed.), Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-81518-1</bdi></cite></li>

<li><cite id="CITEREFIsing1925" class="citation cs2">Ising, E. (1925), "Beitrag zur Theorie des Ferromagnetismus", Z. Phys., 31 (1): 253–258, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1925ZPhy...31..253I">1925ZPhy...31..253I</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02980577">10.1007/BF02980577</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122157319">122157319</a></cite></li>

<li><cite id="CITEREFItzyksonDrouffe1989" class="citation cs2">Itzykson, Claude; Drouffe, Jean-Michel (1989), Théorie statistique des champs, Volume 1, Savoirs actuels (<a href="CNRS" class="mw-redirect" title="CNRS">CNRS</a>), EDP Sciences Editions, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-2-86883-360-0</bdi></cite></li>

<li><cite id="CITEREFItzyksonDrouffe1989" class="citation cs2">Itzykson, Claude; Drouffe, Jean-Michel (1989), Statistical field theory, Volume 1: From Brownian motion to renormalization and lattice gauge theory, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-40805-9</bdi></cite></li>

<li><cite id="CITEREFFriedliVelenik2017" class="citation book cs1">Friedli, S.; Velenik, Y. (2017). <a rel="nofollow" class="external text" href="http://www.unige.ch/math/folks/velenik/smbook/index.html">Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction</a>. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781107184824</bdi>.</cite></li>

<li>Ross Kindermann and J. Laurie Snell (1980), <a rel="nofollow" class="external text" href="https://www.ams.org/online_bks/conm1/">Markov Random Fields and Their Applications</a>. American Mathematical Society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-3381-2</bdi>.</li>

<li><a href="Hagen_Kleinert" title="Hagen Kleinert">Kleinert, H</a> (1989), Gauge Fields in Condensed Matter, Vol. I, "Superflow and Vortex Lines", pp. 1–742, Vol. II, "Stresses and Defects", pp. 743–1456, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100113041810/http://worldscibooks.com/physics/0356.html">World Scientific (Singapore)</a>; Paperback <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9971-5-0210-0</bdi> (also available online: <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html">Vol. I</a> and <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html">Vol. II</a>)</li>

<li><a href="Hagen_Kleinert" title="Hagen Kleinert">Kleinert, H</a> and Schulte-Frohlinde, V (2001), Critical Properties of φ4-Theories, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080226151023/http://www.worldscibooks.com/physics/4733.html">World Scientific (Singapore)</a>; Paperback <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-02-4658-7</bdi> (also available <a rel="nofollow" class="external text" href="http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=6">online</a>)</li>

<li><cite id="CITEREFLenz1920" class="citation cs2"><a href="Wilhelm_Lenz" title="Wilhelm Lenz">Lenz, W.</a> (1920), "Beiträge zum Verständnis der magnetischen Eigenschaften in festen Körpern", Physikalische Zeitschrift, 21: 613–615</cite></li>

<li>Barry M. McCoy and Tai Tsun Wu (1973), The Two-Dimensional Ising Model. Harvard University Press, Cambridge Massachusetts, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-674-91440-6</bdi></li>

<li><cite id="CITEREFMontrollPottsWard1963" class="citation cs2">Montroll, Elliott W.; Potts, Renfrey B.; <a href="John_Clive_Ward" title="John Clive Ward">Ward, John C.</a> (1963), <a rel="nofollow" class="external text" href="https://archive.today/20130112095848/http://link.aip.org/link/?JMAPAQ/4/308/1">"Correlations and spontaneous magnetization of the two-dimensional Ising model"</a>, <a href="Journal_of_Mathematical_Physics" title="Journal of Mathematical Physics">Journal of Mathematical Physics</a>, 4 (2): 308–322, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963JMP.....4..308M">1963JMP.....4..308M</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1703955">10.1063/1.1703955</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0148406">0148406</a>, archived from <a rel="nofollow" class="external text" href="http://link.aip.org/link/?JMAPAQ%2F4%2F308%2F1">the original</a> on 2013-01-12, retrieved 2009-10-25</cite></li>

<li><cite id="CITEREFOnsager1944" class="citation cs2"><a href="Lars_Onsager" title="Lars Onsager">Onsager, Lars</a> (1944), "Crystal statistics. I. A two-dimensional model with an order-disorder transition", Physical Review, Series II, 65 (3–4): 117–149, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1944PhRv...65..117O">1944PhRv...65..117O</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.65.117">10.1103/PhysRev.65.117</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0010315">0010315</a></cite></li>

<li><cite id="CITEREFOnsager1949" class="citation cs2"><a href="Lars_Onsager" title="Lars Onsager">Onsager, Lars</a> (1949), "Discussion", Supplemento al Nuovo Cimento, 6: 261</cite></li>

<li>John Palmer (2007), Planar Ising Correlations. Birkhäuser, Boston, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4248-8</bdi>.</li>

<li><cite id="CITEREFIstrail2000" class="citation cs2">Istrail, Sorin (2000), <a rel="nofollow" class="external text" href="http://www.cs.brown.edu/~sorin/pdfs/Ising-paper.pdf">"Statistical mechanics, three-dimensionality and NP-completeness. I. Universality of intractability for the partition function of the Ising model across non-planar surfaces (extended abstract)"</a> (PDF), Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM, pp. 87–96, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F335305.335316">10.1145/335305.335316</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1581131840</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2114521">2114521</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7944336">7944336</a></cite></li>

<li><cite id="CITEREFYang1952" class="citation cs2"><a href="C._N._Yang" class="mw-redirect" title="C. N. Yang">Yang, C. N.</a> (1952), "The spontaneous magnetization of a two-dimensional Ising model", Physical Review, Series II, 85 (5): 808–816, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952PhRv...85..808Y">1952PhRv...85..808Y</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.85.808">10.1103/PhysRev.85.808</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0051740">0051740</a></cite></li>

<li><cite id="CITEREFGlasser1970" class="citation cs2">Glasser, M. L. (1970), "Exact Partition Function for the Two-Dimensional Ising Model", American Journal of Physics, 38 (8): 1033–1036, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1970AmJPh..38.1033G">1970AmJPh..38.1033G</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1976530">10.1119/1.1976530</a></cite></li>

<li><cite id="CITEREFJellito1979" class="citation cs2">Jellito, R. J. (1979), "The Ising Model on a Closed Cayley Tree", Physica, 99A (1): 268–280, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979PhyA...99..268J">1979PhyA...99..268J</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0378-4371%2879%2990134-1">10.1016/0378-4371(79)90134-1</a></cite></li>

<li><cite id="CITEREFKrizanBarthGlasser1983" class="citation cs2">Krizan, J. E.; <a href="Peter_F._Barth" title="Peter F. Barth">Barth, P. F.</a>; Glasser, M.L. (1983), "Exact Phase Transitions for the Ising Model on the Closed Cayley Tree", Physica, 119A, North-Holland Publishing Co.: 230–242, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0378-4371%2883%2990157-7">10.1016/0378-4371(83)90157-7</a></cite></li>

<li><cite id="CITEREFGlasserGoldberg1983" class="citation cs2">Glasser, M. L.; Goldberg, M. (1983), "The Ising model on a closed Cayley tree", Physica, 117A (2): 670–672, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1983PhyA..117..670G">1983PhyA..117..670G</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0378-4371%2883%2990138-3">10.1016/0378-4371(83)90138-3</a></cite></li>

<li><cite id="CITEREFSüzen2014" class="citation cs2">Süzen, Mehmet (2014), "Effective ergodicity in single-spin-flip dynamics", Physical Review E, 90 (3): 032141, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1405.4497">1405.4497</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PhRvE..90c2141S">2014PhRvE..90c2141S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.90.032141">10.1103/PhysRevE.90.032141</a>, <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25314429">25314429</a></cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://web.mit.edu/redingtn/www/netadv/Xising.html">Ising model at The Net Advance of Physics</a></li>

<li><a href="Barry_Arthur_Cipra" title="Barry Arthur Cipra">Barry Arthur Cipra</a>, "The Ising model is <a href="NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>", <a href="SIAM_News" class="mw-redirect" title="SIAM News">SIAM News</a>, Vol. 33, No. 6; <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070926223950/http://www.siam.org/pdf/news/654.pdf">online edition (.pdf)</a></li>

<li><a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/IsingModel.html">Science World article on the Ising Model</a></li>

<li><a rel="nofollow" class="external text" href="http://physics.ucsc.edu/~peter/ising/ising.html">A dynamical 2D Ising java applet by UCSC</a></li>

<li><a rel="nofollow" class="external text" href="https://sites.google.com/view/chremos-group/applets/ising-model">A dynamical 2D Ising java applet</a></li>

<li><a rel="nofollow" class="external text" href="http://www.physics.uci.edu/~etolleru/IsingApplet/IsingApplet.html">A larger/more complicated 2D Ising java applet</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201125045940/http://www.physics.uci.edu/~etolleru/IsingApplet/IsingApplet.html">Archived</a> 2020-11-25 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="https://www.complexity-explorables.org/explorables/i-sing-well-tempered/">“I sing well-tempered” The Ising Model: A simple model for critical behavior in a system of spins</a> by Dirk Brockman, is an interactive simulation that allows users to export the working code to a presentation slide</li>

<li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/IsingModel/">Ising Model simulation</a> by Enrique Zeleny, the <a href="Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a></li>

<li><a rel="nofollow" class="external text" href="http://ibiblio.org/e-notes/Perc/contents.htm">Phase transitions on lattices</a></li>

<li><a rel="nofollow" class="external text" href="http://www.sandia.gov/media/NewsRel/NR2000/ising.htm">Three-dimensional proof for Ising Model impossible, Sandia researcher claims</a></li>

<li><a rel="nofollow" class="external text" href="http://isingspinwebgl.com">Interactive Monte Carlo simulation of the Ising, XY and Heisenberg models with 3D graphics (requires WebGL compatible browser)</a></li>

<li><a rel="nofollow" class="external text" href="https://github.com/AmazaspShumik/BayesianML-MCMC/blob/master/Gibbs%20Ising%20Model/GibbsIsingModel.m">Ising Model code </a>, <a rel="nofollow" class="external text" href="https://github.com/AmazaspShumik/BayesianML-MCMC/blob/master/Gibbs%20Ising%20Model/imageDenoisingExample.m">image denoising example with Ising Model</a></li>

<li><a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/tong/statphys/five.pdf">David Tong's Lecture Notes </a> provide a good introduction</li>

<li><a rel="nofollow" class="external text" href="https://www.quantamagazine.org/the-cartoon-picture-of-magnets-that-has-transformed-science-20200624/">The Cartoon Picture of Magnets That Has Transformed Science</a> - <a href="Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a> article about Ising model</li>

<li>Simulation of the 2-dimensional Ising model in Julia: <a rel="nofollow" class="external free" href="https://github.com/cossio/SquareIsingModel.jl">https://github.com/cossio/SquareIsingModel.jl</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Stochastic_processes496" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Moran_process" title="Moran process">Moran process</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul><li><a href="Compound_Poisson_process" title="Compound Poisson process">Compound</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

<li><a href="Quasimartingale" title="Quasimartingale">Quasimartingale</a></li>

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

<li><a href="Semimartingale" title="Semimartingale">Semimartingale</a></li>

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

<li><a href="Superprocess" title="Superprocess">Superprocess</a></li>

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

<li><a href="Boolean_network" title="Boolean network">Boolean network</a></li></ul></li>

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

<li><a href="Random_graph" title="Random graph">Random graph</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li>

<li><a href="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1076349#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1067" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1076349#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1067" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://id.worldcat.org/fast/979762">FAST</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4127615-2">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85068376">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119821081">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119821081">BnF data</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007563055005171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://www.idref.fr/027872092">IdRef</a></li><li><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/d55bdfa6-8776-457f-816e-c70bbcdf412a">Yale LUX</a></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-06" href="https://en.wikipedia.org/wiki/?title=Ising_model&oldid=1304520728">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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