SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Partition function (mathematics)</title>

</head>

Partition function (mathematics)

</h1>

</header>

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

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

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

</style><div role="note" class="hatnote navigation-not-searchable">For the partition function in number theory, see <a href="Partition_function_(number_theory)" title="Partition function (number theory)">Partition function (number theory)</a>.</div>

The partition function or configuration integral, as used in <a href="Probability_theory" title="Probability theory">probability theory</a>, <a href="Information_theory" title="Information theory">information theory</a> and <a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>, is a generalization of the definition of a <a href="Partition_function_in_statistical_mechanics" class="mw-redirect" title="Partition function in statistical mechanics">partition function in statistical mechanics</a>. It is a special case of a <a href="Normalizing_constant" title="Normalizing constant">normalizing constant</a> in probability theory, for the <a href="Boltzmann_distribution" title="Boltzmann distribution">Boltzmann distribution</a>. The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated <a href="Probability_measure" title="Probability measure">probability measure</a>, the <a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a>, has the <a href="Markov_property" title="Markov property">Markov property</a>. This means that the partition function occurs not only in physical systems with translation symmetry, but also in such varied settings as neural networks (the <a href="Hopfield_network" title="Hopfield network">Hopfield network</a>), and applications such as <a href="Genomics" title="Genomics">genomics</a>, <a href="Corpus_linguistics" title="Corpus linguistics">corpus linguistics</a> and <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, which employ <a href="Markov_network" class="mw-redirect" title="Markov network">Markov networks</a>, and <a href="Markov_logic_network" title="Markov logic network">Markov logic networks</a>. The Gibbs measure is also the unique measure that has the property of maximizing the <a href="Entropy_(general_concept)" class="mw-redirect" title="Entropy (general concept)">entropy</a> for a fixed expectation value of the energy; this underlies the appearance of the partition function in <a href="Maximum_entropy_method" class="mw-redirect" title="Maximum entropy method">maximum entropy methods</a> and the algorithms derived therefrom.

The partition function ties together many different concepts, and thus offers a general framework in which many different kinds of quantities may be calculated. In particular, it shows how to calculate <a href="Expectation_value" class="mw-redirect" title="Expectation value">expectation values</a> and <a href="Green's_function" title="Green's function">Green's functions</a>, forming a bridge to <a href="Fredholm_theory" title="Fredholm theory">Fredholm theory</a>. It also provides a natural setting for the <a href="Information_geometry" title="Information geometry">information geometry</a> approach to information theory, where the <a href="Fisher_information_metric" title="Fisher information metric">Fisher information metric</a> can be understood to be a <a href="Correlation_function" title="Correlation function">correlation function</a> derived from the partition function; it happens to define a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>.

When the setting for random variables is on <a href="Complex_projective_space" title="Complex projective space">complex projective space</a> or <a href="Projective_Hilbert_space" title="Projective Hilbert space">projective Hilbert space</a>, geometrized with the <a href="Fubini%E2%80%93Study_metric" title="Fubini–Study metric">Fubini–Study metric</a>, the theory of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and more generally <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> results. In these theories, the partition function is heavily exploited in the <a href="Path_integral_formulation" title="Path integral formulation">path integral formulation</a>, with great success, leading to many formulas nearly identical to those reviewed here. However, because the underlying measure space is complex-valued, as opposed to the real-valued <a href="Simplex" title="Simplex">simplex</a> of probability theory, an extra factor of i appears in many formulas. Tracking this factor is troublesome, and is not done here. This article focuses primarily on classical probability theory, where the sum of probabilities total to one.

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

Given a set of <a href="Random_variable" title="Random variable">random variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> taking on values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">, and some sort of <a href="Scalar_potential" title="Scalar potential">potential function</a> or <a href="Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x_{1},x_{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(x_{1},x_{2},\dots )}</annotation>

</semantics>

</math><img src="./4505e24ce28834406fb1b2df625a7e9a2e945e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.432ex; height:2.843ex;" alt="{\displaystyle H(x_{1},x_{2},\dots )}" loading="lazy">, the partition function is defined as

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\sum _{x_{i}}\exp \left(-\beta H(x_{1},x_{2},\dots )\right)}</annotation>

</semantics>

</math>

The function H is understood to be a real-valued function on the space of states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{1},X_{2},\dots \}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{X_{1},X_{2},\dots \}}</annotation>

</semantics>

</math><img src="./c38df3fea84600a5a1571ce5edac667bc37d08f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.073ex; height:2.843ex;" alt="{\displaystyle \{X_{1},X_{2},\dots \}}" loading="lazy">, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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 a real-valued free parameter (conventionally, the <a href="Inverse_temperature" class="mw-redirect" title="Inverse temperature">inverse temperature</a>). The sum over the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> is understood to be a sum over all possible values that each of the random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> may take. Thus, the sum is to be replaced by an <a href="Integral" title="Integral">integral</a> when the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> are continuous, rather than discrete. Thus, one writes

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\int \exp \left(-\beta H(x_{1},x_{2},\dots )\right)\,dx_{1}\,dx_{2}\cdots }</annotation>

</semantics>

</math>

for the case of continuously-varying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy">.

When H is an <a href="Observable" title="Observable">observable</a>, such as a finite-dimensional <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> or an infinite-dimensional <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a> or element of a <a href="C-star_algebra" class="mw-redirect" title="C-star algebra">C-star algebra</a>, it is common to express the summation as a <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a>, so that

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\operatorname {tr} \left(\exp \left(-\beta H\right)\right)}</annotation>

</semantics>

</math>

When H is infinite-dimensional, then, for the above notation to be valid, the argument must be <a href="Trace_class" title="Trace class">trace class</a>, that is, of a form such that the summation exists and is bounded.

The number of variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> need not be <a href="Countable" class="mw-redirect" title="Countable">countable</a>, in which case the sums are to be replaced by <a href="Functional_integral" class="mw-redirect" title="Functional integral">functional integrals</a>. Although there are many notations for functional integrals, a common one would be

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z=\int {\mathcal {D}}\varphi \exp \left(-\beta H[\varphi ]\right)}</annotation>

</semantics>

</math>

Such is the case for the <a href="Partition_function_in_quantum_field_theory" class="mw-redirect" title="Partition function in quantum field theory">partition function in quantum field theory</a>.

A common, useful modification to the partition function is to introduce auxiliary functions. This allows, for example, the partition function to be used as a <a href="Generating_function" title="Generating function">generating function</a> for <a href="Correlation_function" title="Correlation function">correlation functions</a>. This is discussed in greater detail below.

<div class="mw-heading mw-heading2"><h2 id="The_parameter_β">The parameter β</h2></div>

The role or meaning of the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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"> can be understood in a variety of different ways. In classical thermodynamics, it is an <a href="Inverse_temperature" class="mw-redirect" title="Inverse temperature">inverse temperature</a>. More generally, one would say that it is the variable that is <a href="Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">conjugate</a> to some (arbitrary) function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"> of the random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. The word conjugate here is used in the sense of conjugate <a href="Generalized_coordinates" title="Generalized coordinates">generalized coordinates</a> in <a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>, thus, properly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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 a <a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a>. It is not uncommonly called the <a href="Generalized_force" class="mw-redirect" title="Generalized force">generalized force</a>. All of these concepts have in common the idea that one value is meant to be kept fixed, as others, interconnected in some complicated way, are allowed to vary. In the current case, the value to be kept fixed is the <a href="Expectation_value" class="mw-redirect" title="Expectation value">expectation value</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy">, even as many different <a href="Probability_distribution" title="Probability distribution">probability distributions</a> can give rise to exactly this same (fixed) value.

For the general case, one considers a set of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{H_{k}(x_{1},\dots )\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{H_{k}(x_{1},\dots )\}}</annotation>

</semantics>

</math><img src="./b26d737aeffec3bc3fae75b38b81d47d799ed410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.295ex; height:2.843ex;" alt="{\displaystyle \{H_{k}(x_{1},\dots )\}}" loading="lazy"> that each depend on the random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy">. These functions are chosen because one wants to hold their expectation values constant, for one reason or another. To constrain the expectation values in this way, one applies the method of <a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multipliers</a>. In the general case, <a href="Maximum_entropy_method" class="mw-redirect" title="Maximum entropy method">maximum entropy methods</a> illustrate the manner in which this is done.

Some specific examples are in order. In basic thermodynamics problems, when using the <a href="Canonical_ensemble" title="Canonical ensemble">canonical ensemble</a>, the use of just one parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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"> reflects the fact that there is only one expectation value that must be held constant: the <a href="Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> (due to <a href="Conservation_of_energy" title="Conservation of energy">conservation of energy</a>). For chemistry problems involving chemical reactions, the <a href="Grand_canonical_ensemble" title="Grand canonical ensemble">grand canonical ensemble</a> provides the appropriate foundation, and there are two Lagrange multipliers. One is to hold the energy constant, and another, the <a href="Fugacity" title="Fugacity">fugacity</a>, is to hold the particle count constant (as chemical reactions involve the recombination of a fixed number of atoms).

For the general case, one has

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\sum _{x_{i}}\exp \left(-\sum _{k}\beta _{k}H_{k}(x_{i})\right)}</annotation>

</semantics>

</math>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =(\beta _{1},\beta _{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta =(\beta _{1},\beta _{2},\dots )}</annotation>

</semantics>

</math><img src="./575ee68cec4b8022774c2b1726c3694d8de90d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.771ex; height:2.843ex;" alt="{\displaystyle \beta =(\beta _{1},\beta _{2},\dots )}" loading="lazy"> a point in a space.

For a collection of observables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4c8af69c8cae9ee6ba302b6b3f4b0618ac08e427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle H_{k}}" loading="lazy">, one would write

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\operatorname {tr} \left[\,\exp \left(-\sum _{k}\beta _{k}H_{k}\right)\right]}</annotation>

</semantics>

</math>

As before, it is presumed that the argument of tr is <a href="Trace_class" title="Trace class">trace class</a>.

The corresponding <a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a> then provides a probability distribution such that the expectation value of each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4c8af69c8cae9ee6ba302b6b3f4b0618ac08e427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle H_{k}}" loading="lazy"> is a fixed value. More precisely, one has

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial \beta _{k}}}\left(-\log Z\right)=\langle H_{k}\rangle =\mathrm {E} \left[H_{k}\right]}</annotation>

</semantics>

</math>

with the angle brackets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle H_{k}\rangle }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e8f197e2d147c77fcef318c9096c7ff4811ad5f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.843ex;" alt="{\displaystyle \langle H_{k}\rangle }" loading="lazy"> denoting the expected value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4c8af69c8cae9ee6ba302b6b3f4b0618ac08e427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle H_{k}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [\,\cdot \,]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\,\cdot \,]}</annotation>

</semantics>

</math><img src="./6df9b3afe62533b789f25a205fe97c1d20be5d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.298ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\,\cdot \,]}" loading="lazy"> being a common alternative notation. A precise definition of this expectation value is given below.

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

</mstyle>

</mrow>

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

</semantics>

</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 commonly taken to be real, it need not be, in general; this is discussed in the section <a href="#Normalization">Normalization</a> below. The values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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"> can be understood to be the coordinates of points in a space; this space is in fact a <a href="Manifold" title="Manifold">manifold</a>, as sketched below. The study of these spaces as manifolds constitutes the field of <a href="Information_geometry" title="Information geometry">information geometry</a>.

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

The potential function itself commonly takes the form of a sum:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(x_{1},x_{2},\dots )=\sum _{s}V(s)\,}</annotation>

</semantics>

</math>

where the sum over s is a sum over some subset of the <a href="Power_set" title="Power set">power set</a> P(X) of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\lbrace x_{1},x_{2},\dots \rbrace }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\lbrace x_{1},x_{2},\dots \rbrace }</annotation>

</semantics>

</math><img src="./54a7a96af0dbb18f2d762f737f72f030b43407b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.962ex; height:2.843ex;" alt="{\displaystyle X=\lbrace x_{1},x_{2},\dots \rbrace }" loading="lazy">. For example, in <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, such as the <a href="Ising_model" title="Ising model">Ising model</a>, the sum is over pairs of nearest neighbors. In probability theory, such as <a href="Markov_networks" class="mw-redirect" title="Markov networks">Markov networks</a>, the sum might be over the <a href="Clique_(graph_theory)" title="Clique (graph theory)">cliques</a> of a graph; so, for the Ising model and other <a href="Lattice_model_(physics)" title="Lattice model (physics)">lattice models</a>, the maximal cliques are edges.

The fact that the potential function can be written as a sum usually reflects the fact that it is invariant under the <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of a <a href="Group_(mathematics)" title="Group (mathematics)">group symmetry</a>, such as <a href="Translational_invariance" class="mw-redirect" title="Translational invariance">translational invariance</a>. Such symmetries can be discrete or continuous; they materialize in the <a href="Correlation_function" title="Correlation function">correlation functions</a> for the random variables (discussed below). Thus a symmetry in the Hamiltonian becomes a symmetry of the correlation function (and vice versa).

This symmetry has a critically important interpretation in probability theory: it implies that the <a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a> has the <a href="Markov_property" title="Markov property">Markov property</a>; that is, it is independent of the random variables in a certain way, or, equivalently, the measure is identical on the <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> of the symmetry. This leads to the widespread appearance of the partition function in problems with the Markov property, such as <a href="Hopfield_network" title="Hopfield network">Hopfield networks</a>.

<div class="mw-heading mw-heading2"><h2 id="As_a_measure">As a measure</h2></div>

The value of the expression

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left(-\beta H(x_{1},x_{2},\dots )\right)}</annotation>

</semantics>

</math>

can be interpreted as a likelihood that a specific <a href="Configuration_space_(physics)" title="Configuration space (physics)">configuration</a> of values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\dots )}</annotation>

</semantics>

</math><img src="./0d1fabaef4eaa2225bb42a4573912d997ef34c4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\dots )}" loading="lazy"> occurs in the system. Thus, given a specific configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\dots )}</annotation>

</semantics>

</math><img src="./0d1fabaef4eaa2225bb42a4573912d997ef34c4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\dots )}" loading="lazy">,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x_{1},x_{2},\dots )={\frac {1}{Z(\beta )}}\exp \left(-\beta H(x_{1},x_{2},\dots )\right)}</annotation>

</semantics>

</math>

is the <a href="Probability_density_function" title="Probability density function">probability</a> of the configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\dots )}</annotation>

</semantics>

</math><img src="./0d1fabaef4eaa2225bb42a4573912d997ef34c4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\dots )}" loading="lazy"> occurring in the system, which is now properly normalized so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq P(x_{1},x_{2},\dots )\leq 1}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq P(x_{1},x_{2},\dots )\leq 1}</annotation>

</semantics>

</math><img src="./1869c52b37f4736c0e507bfb100b3efe49ed73d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.636ex; height:2.843ex;" alt="{\displaystyle 0\leq P(x_{1},x_{2},\dots )\leq 1}" loading="lazy">, and such that the sum over all configurations totals to one. As such, the partition function can be understood to provide a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> (a <a href="Probability_measure" title="Probability measure">probability measure</a>) on the <a href="Probability_space" title="Probability space">probability space</a>; formally, it is called the <a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a>. It generalizes the narrower concepts of the <a href="Grand_canonical_ensemble" title="Grand canonical ensemble">grand canonical ensemble</a> and <a href="Canonical_ensemble" title="Canonical ensemble">canonical ensemble</a> in statistical mechanics.

There exists at least one configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\dots )}</annotation>

</semantics>

</math><img src="./0d1fabaef4eaa2225bb42a4573912d997ef34c4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\dots )}" loading="lazy"> for which the probability is maximized; this configuration is conventionally called the <a href="Ground_state" title="Ground state">ground state</a>. If the configuration is unique, the ground state is said to be non-degenerate, and the system is said to be <a href="Ergodic" class="mw-redirect" title="Ergodic">ergodic</a>; otherwise the ground state is degenerate. The ground state may or may not commute with the generators of the symmetry; if commutes, it is said to be an <a href="Invariant_measure" title="Invariant measure">invariant measure</a>. When it does not commute, the symmetry is said to be <a href="Spontaneously_broken" class="mw-redirect" title="Spontaneously broken">spontaneously broken</a>.

Conditions under which a ground state exists and is unique are given by the <a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a>; these conditions are commonly used to justify the use of the Gibbs measure in maximum-entropy problems.

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

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

</mstyle>

</mrow>

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

</semantics>

</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"> depend on the <a href="Mathematical_space" class="mw-redirect" title="Mathematical space">mathematical space</a> over which the random field varies. Thus, real-valued random fields take values on a <a href="Simplex" title="Simplex">simplex</a>: this is the geometrical way of saying that the sum of probabilities must total to one. For quantum mechanics, the random variables range over <a href="Complex_projective_space" title="Complex projective space">complex projective space</a> (or complex-valued <a href="Projective_Hilbert_space" title="Projective Hilbert space">projective Hilbert space</a>), where the random variables are interpreted as <a href="Probability_amplitude" title="Probability amplitude">probability amplitudes</a>. The emphasis here is on the word projective, as the amplitudes are still normalized to one. The normalization for the potential function is the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> for the appropriate mathematical space: it is 1 for ordinary probabilities, and i for Hilbert space; thus, in <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, one sees <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle itH}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9a3405b7bc569869354be33cc2c73d536f4495e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.706ex; height:2.176ex;" alt="{\displaystyle itH}" loading="lazy"> in the exponential, rather than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f2d18e052b0e2ef4884adecb9690bfe75cce6cb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.396ex; height:2.509ex;" alt="{\displaystyle \beta H}" loading="lazy">. The partition function is very heavily exploited in the <a href="Path_integral_formulation" title="Path integral formulation">path integral formulation</a> of quantum field theory, to great effect. The theory there is very nearly identical to that presented here, aside from this difference, and the fact that it is usually formulated on four-dimensional space-time, rather than in a general way.

<div class="mw-heading mw-heading2"><h2 id="Expectation_values">Expectation values</h2></div>

The partition function is commonly used as a <a href="Probability-generating_function" title="Probability-generating function">probability-generating function</a> for <a href="Expectation_value" class="mw-redirect" title="Expectation value">expectation values</a> of various functions of the random variables. So, for example, taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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"> as an adjustable parameter, then the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(Z(\beta ))}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b44b9fa92c645183b11b786f8996d965427487fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.603ex; height:2.843ex;" alt="{\displaystyle \log(Z(\beta ))}" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [H]=\langle H\rangle =-{\frac {\partial \log(Z(\beta ))}{\partial \beta }}}</annotation>

</semantics>

</math>

gives the average (expectation value) of H. In physics, this would be called the average <a href="Energy" title="Energy">energy</a> of the system.

Given the definition of the probability measure above, the expectation value of any function f of the random variables X may now be written as expected: so, for discrete-valued X, one writes

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle f\rangle &=\sum _{x_{i}}f(x_{1},x_{2},\dots )P(x_{1},x_{2},\dots )\\&={\frac {1}{Z(\beta )}}\sum _{x_{i}}f(x_{1},x_{2},\dots )\exp \left(-\beta H(x_{1},x_{2},\dots )\right)\end{aligned}}}</annotation>

</semantics>

</math>

The above notation makes sense for a finite number of discrete random variables. In more general settings, the summations should be replaced with integrals over a <a href="Probability_space" title="Probability space">probability space</a>.

Thus, for example, the <a href="Entropy_(general_concept)" class="mw-redirect" title="Entropy (general concept)">entropy</a> is given by

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S&=-k_{\text{B}}\langle \ln P\rangle \\[1ex]&=-k_{\text{B}}\sum _{x_{i}}P(x_{1},x_{2},\dots )\ln P(x_{1},x_{2},\dots )\\&=k_{\text{B}}\left(\beta \langle H\rangle +\log Z(\beta )\right)\end{aligned}}}</annotation>

</semantics>

</math>

The Gibbs measure is the unique statistical distribution that maximizes the entropy for a fixed expectation value of the energy; this underlies its use in <a href="Maximum_entropy_method" class="mw-redirect" title="Maximum entropy method">maximum entropy methods</a>.

<div class="mw-heading mw-heading2"><h2 id="Information_geometry">Information geometry</h2></div>

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

</mstyle>

</mrow>

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

</semantics>

</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"> can be understood to form a space, and specifically, a <a href="Manifold" title="Manifold">manifold</a>. Thus, it is reasonable to ask about the structure of this manifold; this is the task of <a href="Information_geometry" title="Information geometry">information geometry</a>.

Multiple derivatives with regard to the Lagrange multipliers gives rise to a positive semi-definite <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{ij}(\beta )={\frac {\partial ^{2}}{\partial \beta ^{i}\partial \beta ^{j}}}\left(-\log Z(\beta )\right)=\langle \left(H_{i}-\langle H_{i}\rangle \right)\left(H_{j}-\langle H_{j}\rangle \right)\rangle }</annotation>

</semantics>

</math>

This matrix is positive semi-definite, and may be interpreted as a <a href="Metric_tensor" title="Metric tensor">metric tensor</a>, specifically, a <a href="Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a>. Equipping the space of Lagrange multipliers with a metric in this way turns it into a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>.<a href="#cite_note-1">[1]</a> The study of such manifolds is referred to as <a href="Information_geometry" title="Information geometry">information geometry</a>; the metric above is the <a href="Fisher_information_metric" title="Fisher information metric">Fisher information metric</a>. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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"> serves as a coordinate on the manifold. It is interesting to compare the above definition to the simpler <a href="Fisher_information" title="Fisher information">Fisher information</a>, from which it is inspired.

That the above defines the Fisher information metric can be readily seen by explicitly substituting for the expectation value:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{ij}(\beta )&=\left\langle \left(H_{i}-\left\langle H_{i}\right\rangle \right)\left(H_{j}-\left\langle H_{j}\right\rangle \right)\right\rangle \\&=\sum _{x}P(x)\left(H_{i}-\left\langle H_{i}\right\rangle \right)\left(H_{j}-\left\langle H_{j}\right\rangle \right)\\&=\sum _{x}P(x)\left(H_{i}+{\frac {\partial \log Z}{\partial \beta _{i}}}\right)\left(H_{j}+{\frac {\partial \log Z}{\partial \beta _{j}}}\right)\\&=\sum _{x}P(x){\frac {\partial \log P(x)}{\partial \beta ^{i}}}{\frac {\partial \log P(x)}{\partial \beta ^{j}}}\\\end{aligned}}}</annotation>

</semantics>

</math>

where we've written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./89833156eff2c51bfb8750db3306a0544ce34e14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{1},x_{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x_{1},x_{2},\dots )}</annotation>

</semantics>

</math><img src="./f958cab38cfc31d33c6395d1972d45c9a7ba29e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.114ex; height:2.843ex;" alt="{\displaystyle P(x_{1},x_{2},\dots )}" loading="lazy"> and the summation is understood to be over all values of all random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./33c25229c6989c235f9cbb7908331f6d01d0abfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}" loading="lazy">. For continuous-valued random variables, the summations are replaced by integrals, of course.

Curiously, the <a href="Fisher_information_metric" title="Fisher information metric">Fisher information metric</a> can also be understood as the flat-space <a href="Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>, after appropriate change of variables, as described in the main article on it. When the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</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"> are complex-valued, the resulting metric is the <a href="Fubini%E2%80%93Study_metric" title="Fubini–Study metric">Fubini–Study metric</a>. When written in terms of <a href="Mixed_state_(physics)" class="mw-redirect" title="Mixed state (physics)">mixed states</a>, instead of <a href="Pure_state" class="mw-redirect" title="Pure state">pure states</a>, it is known as the <a href="Bures_metric" title="Bures metric">Bures metric</a>.

<div class="mw-heading mw-heading2"><h2 id="Correlation_functions">Correlation functions</h2></div>

By introducing artificial auxiliary functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a99023826aca8b6316815468d3dcf4a512f383eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.509ex;" alt="{\displaystyle J_{k}}" loading="lazy"> into the partition function, it can then be used to obtain the expectation value of the random variables. Thus, for example, by writing

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Z(\beta ,J)&=Z(\beta ,J_{1},J_{2},\dots )\\&=\sum _{x_{i}}\exp \left(-\beta H(x_{1},x_{2},\dots )+\sum _{n}J_{n}x_{n}\right)\end{aligned}}}</annotation>

</semantics>

</math>

one then has

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [x_{k}]=\langle x_{k}\rangle =\left.{\frac {\partial }{\partial J_{k}}}\log Z(\beta ,J)\right|_{J=0}}</annotation>

</semantics>

</math>

as the expectation value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy">. In the <a href="Path_integral_formulation" title="Path integral formulation">path integral formulation</a> of <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, these auxiliary functions are commonly referred to as <a href="Source_field" title="Source field">source fields</a>.

Multiple differentiations lead to the <a href="Ursell_function" title="Ursell function">connected correlation functions</a> of the random variables. Thus the correlation function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x_{j},x_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(x_{j},x_{k})}</annotation>

</semantics>

</math><img src="./e98c929115d0c2ae07db564a9041384a8f7d6cef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.267ex; height:3.009ex;" alt="{\displaystyle C(x_{j},x_{k})}" loading="lazy"> between variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"> is given by:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(x_{j},x_{k})=\left.{\frac {\partial }{\partial J_{j}}}{\frac {\partial }{\partial J_{k}}}\log Z(\beta ,J)\right|_{J=0}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Gaussian_integrals">Gaussian integrals</h2></div>

For the case where H can be written as a <a href="Quadratic_form" title="Quadratic form">quadratic form</a> involving a <a href="Differential_operator" title="Differential operator">differential operator</a>, that is, as

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H={\frac {1}{2}}\sum _{n}x_{n}Dx_{n}}</annotation>

</semantics>

</math>

then partition function can be understood to be a sum or <a href="Gaussian_integral" title="Gaussian integral">integral</a> over Gaussians. The correlation function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x_{j},x_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(x_{j},x_{k})}</annotation>

</semantics>

</math><img src="./e98c929115d0c2ae07db564a9041384a8f7d6cef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.267ex; height:3.009ex;" alt="{\displaystyle C(x_{j},x_{k})}" loading="lazy"> can be understood to be the <a href="Green's_function" title="Green's function">Green's function</a> for the differential operator (and generally giving rise to <a href="Fredholm_theory" title="Fredholm theory">Fredholm theory</a>). In the quantum field theory setting, such functions are referred to as <a href="Propagator" title="Propagator">propagators</a>; higher order correlators are called n-point functions; working with them defines the <a href="Effective_action" title="Effective action">effective action</a> of a theory.

When the random variables are anti-commuting <a href="Grassmann_number" title="Grassmann number">Grassmann numbers</a>, then the partition function can be expressed as a determinant of the operator D. This is done by writing it as a <a href="Berezin_integral" title="Berezin integral">Berezin integral</a> (also called Grassmann integral).

<div class="mw-heading mw-heading2"><h2 id="General_properties">General properties</h2></div>

Partition functions are used to discuss <a href="Critical_scaling" class="mw-redirect" title="Critical scaling">critical scaling</a>, <a href="Universality_(dynamical_systems)" title="Universality (dynamical systems)">universality</a> and are subject to the <a href="Renormalization_group" title="Renormalization group">renormalization group</a>.

<ul><li><a href="Exponential_family" title="Exponential family">Exponential family</a></li>

<li><a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">Partition function (statistical mechanics)</a></li>

<li><a href="Partition_problem" title="Partition problem">Partition problem</a></li>

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

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

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

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

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

</style><div class="reflist">

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

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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

</style><cite id="CITEREFCrooks2007" class="citation journal cs1">Crooks, Gavin E. (2007). "Measuring Thermodynamic Length". <a href="Physical_Review_Letters" title="Physical Review Letters">Phys. Rev. Lett.</a> 99 (10): 100602. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.0559">0706.0559</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/2007PhRvL..99j0602C">2007PhRvL..99j0602C</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%2FPhysRevLett.99.100602">10.1103/PhysRevLett.99.100602</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/17930381">17930381</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:7527491">7527491</a>.</cite>

</li>

</ol></div></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-03-17" href="https://en.wikipedia.org/wiki/?title=Partition_function_(mathematics)&oldid=1281017243">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.

</div>

</div></div>

</div>

</main>

</div>

</body></html>