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In [[physics]] and [[probability theory]], '''mean-field theory''' (aka '''MFT''' or rarely '''self-consistent field theory''') studies the behavior of high-dimensional random ([[stochastic]]) models by studying a simpler model that approximates the original by averaging over degrees of freedom. Such models consider many individual components that interact with each other. In MFT, the effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a [[many-body problem]] to a [[one-body problem]].

In [[physics]] and [[probability theory]], '''mean-field theory''' (aka '''MFT''' or rarely '''self-consistent field theory''') studies the behavior of high-dimensional random ([[stochastic]]) models by studying a simpler model that approximates the original by averaging over degrees of freedom. Such models consider many individual components that interact with each other. In MFT, the effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a [[many-body problem]] to a [[one-body problem]].

在物理学和概率论学中，平均场理论(又称 MFT 或自洽场理论)通过研究一个简单的模型来表示高维随机模型的行为，通过对原始模型的自由度取平均来近似得到。这些模型考虑到了许多相互交互的单个组件。在平均场论中，所有个体对任一给定个体的影响都近似于所有个体的平均效应，从而使多体问题降低为一体问题。

 

在物理学和概率论学中，平均场理论(又称 MFT 或罕见的自洽场理论)通过研究一个更简单的模型来研究高维随机(随机)模型的行为，这个模型通过超过自由度的平均来近似原始模型。这些模型考虑了许多相互交互的单个组件。在 MFT 中，所有其他个体对任何给定个体的影响都近似于单一平均效应，从而将多体问题降低为一体问题。

 

 

The main idea of MFT is to replace all interactions to any one body with an average or effective interaction, sometimes called a ''molecular field''.<ref>{{cite book |title=Principles of condensed matter physics |last1=Chaikin |first1=P. M. |last2=Lubensky |first2=T. C. |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-79450-3 |edition=4th print |location=Cambridge}}</ref> This reduces any many-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a lower computational cost.

The main idea of MFT is to replace all interactions to any one body with an average or effective interaction, sometimes called a ''molecular field''.<ref name=":0">{{cite book |title=Principles of condensed matter physics |last1=Chaikin |first1=P. M. |last2=Lubensky |first2=T. C. |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-79450-3 |edition=4th print |location=Cambridge}}</ref> This reduces any many-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a lower computational cost.

MFT 的主要思想是用一个平均的或有效的相互作用，有时称为分子场，来代替任何一个物体的所有相互作用。这就把任何多体问题转化为有效的单体问题。解决 MFT 问题的容易性意味着可以以较低的计算成本获得对系统行为的一些了解。

MFT has since been applied to a wide range of fields outside of physics, including [[statistical inference]], [[graphical models]], [[neuroscience]]<ref name=":1">{{cite journal |last1=Parr |first1=Thomas |last2=Sajid |first2=Noor |last3=Friston |first3=Karl |title=Modules or Mean-Fields? |journal=Entropy |date=2020 |volume=22 |issue=552 |page=552 |doi=10.3390/e22050552 |url=https://res.mdpi.com/d_attachment/entropy/entropy-22-00552/article_deploy/entropy-22-00552.pdf |accessdate=22 May 2020}}</ref>, [[artificial intelligence]], [[epidemic model]]s,<ref name=":2">{{Cite book |url=http://www.cs.toronto.edu/~marbach/ENS/leboudec.pdf |title=Fourth International Conference on the Quantitative Evaluation of Systems (QEST 2007) |last1=Boudec |first1=J. Y. L. |last2=McDonald |first2=D. |last3=Mundinger |first3=J. |year=2007 |isbn=978-0-7695-2883-0 |pages=3 |chapter=A Generic Mean Field Convergence Result for Systems of Interacting Objects |doi=10.1109/QEST.2007.8 |pmc= |pmid= |citeseerx=10.1.1.110.2612|s2cid=15007784 }}</ref> [[queueing theory]],<ref name=":3">{{Cite journal |last1=Baccelli |first1=F. |last2=Karpelevich |first2=F. I. |last3=Kelbert |first3=M. Y. |last4=Puhalskii |first4=A. A. |last5=Rybko |first5=A. N. |last6=Suhov |first6=Y. M. |year=1992 |title=A mean-field limit for a class of queueing networks |journal=Journal of Statistical Physics |volume=66 |issue=3–4 |pages=803 |bibcode=1992JSP....66..803B |doi=10.1007/BF01055703 |pmc= |pmid=|s2cid=120840517 }}</ref> [[Network performance|computer-network performance]] and [[mean-field game theory|game theory]],<ref name=":4">{{Cite journal |last1=Lasry |first1=J. M. |last2=Lions |first2=P. L. |authorlink2=Pierre-Louis Lions |year=2007 |title=Mean field games |journal=Japanese Journal of Mathematics |volume=2 |pages=229–260 |doi=10.1007/s11537-007-0657-8 |pmc= |pmid= |s2cid=1963678 |url=https://basepub.dauphine.fr//bitstream/123456789/2263/1/Cahier_Chaire_2.pdf}}</ref> as in the [[quantal response equilibrium]].

== Origins ==

== Origins ==

The ideas first appeared in physics ([[statistical mechanics]]) in the work of [[Pierre Curie]]<ref>{{Cite journal | last1 = Kadanoff | first1 = L. P. | authorlink1 = Leo Kadanoff| title = More is the Same; Phase Transitions and Mean Field Theories | doi = 10.1007/s10955-009-9814-1 | journal = Journal of Statistical Physics | volume = 137 | issue = 5–6 | pages = 777–797 | year = 2009 | arxiv = 0906.0653| pmid =  | pmc = |bibcode = 2009JSP...137..777K | s2cid = 9074428 }}</ref> and [[Pierre Weiss]] to describe [[phase transitions]].<ref>{{cite journal | title = L'hypothèse du champ moléculaire et la propriété ferromagnétique | first = Pierre | last = Weiss | authorlink = Pierre Weiss | journal = J. Phys. Theor. Appl. | volume = 6 | issue = 1 | year= 1907 | pages= 661–690 | doi = 10.1051/jphystap:019070060066100 | url = http://hal.archives-ouvertes.fr/jpa-00241247/en }}</ref> MFT has been used in the [[Bragg–Williams approximation]], models on [[Bethe lattice]], [[Landau theory]], [[Pierre–Weiss approximation]], [[Flory–Huggins solution theory]], and [[Scheutjens–Fleer theory]].

The ideas first appeared in physics ([[statistical mechanics]]) in the work of [[Pierre Curie]]<ref name=":5">{{Cite journal | last1 = Kadanoff | first1 = L. P. | authorlink1 = Leo Kadanoff| title = More is the Same; Phase Transitions and Mean Field Theories | doi = 10.1007/s10955-009-9814-1 | journal = Journal of Statistical Physics | volume = 137 | issue = 5–6 | pages = 777–797 | year = 2009 | arxiv = 0906.0653| pmid =  | pmc = |bibcode = 2009JSP...137..777K | s2cid = 9074428 }}</ref> and [[Pierre Weiss]] to describe [[phase transitions]].<ref name=":6">{{cite journal | title = L'hypothèse du champ moléculaire et la propriété ferromagnétique | first = Pierre | last = Weiss | authorlink = Pierre Weiss | journal = J. Phys. Theor. Appl. | volume = 6 | issue = 1 | year= 1907 | pages= 661–690 | doi = 10.1051/jphystap:019070060066100 | url = http://hal.archives-ouvertes.fr/jpa-00241247/en }}</ref> MFT has been used in the [[Bragg–Williams approximation]], models on [[Bethe lattice]], [[Landau theory]], [[Pierre–Weiss approximation]], [[Flory–Huggins solution theory]], and [[Scheutjens–Fleer theory]].

The ideas first appeared in physics (statistical mechanics) in the work of Pierre Curie and Pierre Weiss to describe phase transitions. MFT has been used in the Bragg–Williams approximation, models on Bethe lattice, Landau theory, Pierre–Weiss approximation, Flory–Huggins solution theory, and Scheutjens–Fleer theory.

这个想法最早出现在物理学的统计力学中，由 Pierre Curie<ref name=":5" /> 和 Pierre Weiss 描述相变<ref name=":6" />的著作中。平均场论在 Bragg-Williams 近似、 Bethe 晶格模型、 Landau 理论、 Pierre-Weiss 近似、 Flory-Huggins 解理论和 Scheutjens-Fleer 理论中都有应用。

[[Many-body system|Systems]] with many (sometimes infinite) degrees of freedom are generally hard to solve exactly or compute in closed, analytic form, except for some simple cases (e.g. certain Gaussian [[random-field]] theories, the 1D [[Ising model]]). Often combinatorial problems arise that make things like computing the [[Partition function (mathematics)|partition function]] of a system difficult. MFT is an approximation method that often makes the original solvable and open to calculation. Sometimes, MFT gives very accurate approximations.

[[Many-body system|Systems]] with many (sometimes infinite) degrees of freedom are generally hard to solve exactly or compute in closed, analytic form, except for some simple cases (e.g. certain Gaussian [[random-field]] theories, the 1D [[Ising model]]). Often combinatorial problems arise that make things like computing the [[Partition function (mathematics)|partition function]] of a system difficult. MFT is an approximation method that often makes the original solvable and open to calculation. Sometimes, MFT gives very accurate approximations.

Systems with many (sometimes infinite) degrees of freedom are generally hard to solve exactly or compute in closed, analytic form, except for some simple cases (e.g. certain Gaussian random-field theories, the 1D Ising model). Often combinatorial problems arise that make things like computing the partition function of a system difficult. MFT is an approximation method that often makes the original solvable and open to calculation. Sometimes, MFT gives very accurate approximations.

具有许多(有时是无数个)自由度的系统，除了一些简单的情况(例如:高斯随机场理论，一维伊辛模型)，通常难以精确地求解或以封闭的解析形式计算。一个复杂的组合问题的出现使得计算一个系统的配分函数变得困难。平均场论是一种近似方法，它常常使原问题变得可解，易于计算。有时，平均场论可以给出非常精确的近似值。

In [[classical field theory|field theory]], the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field. In this context, MFT can be viewed as the "zeroth-order" expansion of the Hamiltonian in fluctuations.  Physically, this means that an MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean field".

In [[classical field theory|field theory]], the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field. In this context, MFT can be viewed as the "zeroth-order" expansion of the Hamiltonian in fluctuations.  Physically, this means that an MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean field".

Quite often, MFT provides a convenient launch point to studying higher-order fluctuations. For example, when computing the [[Partition function (statistical mechanics)|partition function]], studying the [[combinatorics]] of the interaction terms in the [[Hamiltonian mechanics|Hamiltonian]] can sometimes at best produce [[Perturbation theory|perturbative]] results or [[Feynman diagram]]s that correct the mean-field approximation.

Quite often, MFT provides a convenient launch point to studying higher-order fluctuations. For example, when computing the [[Partition function (statistical mechanics)|partition function]], studying the [[combinatorics]] of the interaction terms in the [[Hamiltonian mechanics|Hamiltonian]] can sometimes at best produce [[Perturbation theory|perturbative]] results or [[Feynman diagram]]s that correct the mean-field approximation.

== Validity ==

== Validity ==

In general, dimensionality plays a strong role in determining whether a mean-field approach will work for any particular problem. There is sometimes a [[critical dimension]], above which MFT is valid and below which it is not.  

In general, dimensionality plays a strong role in determining whether a mean-field approach will work for any particular problem. There is sometimes a [[critical dimension]], above which MFT is valid and below which it is not.  

Heuristically, many interactions are replaced in MFT by one effective interaction. So if the field or particle exhibits many random interactions in the original system, they tend to cancel each other out, so the mean effective interaction and MFT will be more accurate. This is true in cases of high dimensionality, when the Hamiltonian includes long-range forces, or when the particles are extended (e.g. polymers). The [[Ginzburg criterion]] is the formal expression of how fluctuations render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.

Heuristically, many interactions are replaced in MFT by one effective interaction. So if the field or particle exhibits many random interactions in the original system, they tend to cancel each other out, so the mean effective interaction and MFT will be more accurate. This is true in cases of high dimensionality, when the Hamiltonian includes long-range forces, or when the particles are extended (e.g. polymers). The [[Ginzburg criterion]] is the formal expression of how fluctuations render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.

Heuristically, many interactions are replaced in MFT by one effective interaction. So if the field or particle exhibits many random interactions in the original system, they tend to cancel each other out, so the mean effective interaction and MFT will be more accurate. This is true in cases of high dimensionality, when the Hamiltonian includes long-range forces, or when the particles are extended (e.g. polymers). The Ginzburg criterion is the formal expression of how fluctuations render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.

由此，平均场论中的许多相互作用会被一个有效的相互作用所取代。如果场或粒子在原系统中表现出许多随机相互作用，它们往往会相互抵消，从而使平均有效相互作用和平均场论更加精确。这在高维情况下也是成立的，比如当哈密顿量包括远程力时，或者当粒子被扩展时(例如:聚合物)。金兹堡准则是衡量一个近似不好的平均场如何波动的表达式，通常取决于系统中的空间维数。

The formal basis for mean-field theory is the [[Helmholtz free energy#Bogoliubov inequality|Bogoliubov inequality]]. This inequality states that the [[thermodynamic free energy|free energy]] of a system with Hamiltonian

The formal basis for mean-field theory is the [[Helmholtz free energy#Bogoliubov inequality|Bogoliubov inequality]]. This inequality states that the [[thermodynamic free energy|free energy]] of a system with Hamiltonian

平均场理论的形式基础是波格留波夫不等式。这个不等式说明了哈密顿量系统的自由能

平均场理论的形式基础是波格留波夫不等式。这个不等式说明了哈密顿量系统的自由能

  <math>\mathcal{H} = \mathcal{H}_0 + \Delta \mathcal{H}</math>

  <math>\mathcal{H} = \mathcal{H}_0 + \Delta \mathcal{H}</math>

has the following upper bound:

has the following upper bound:

有下面的上界:

上界:

  <math>F \leq F_0 \ \stackrel{\mathrm{def}}{=}\  \langle \mathcal{H} \rangle_0 - T S_0,</math>

  <math>F \leq F_0 \ \stackrel{\mathrm{def}}{=}\  \langle \mathcal{H} \rangle_0 - T S_0,</math>

where <math>S_0</math> is the [[entropy]], and <math>F</math> and <math>F_0</math> are [[Helmholtz free energy|Helmholtz free energies]]. The average is taken over the equilibrium [[Statistical ensemble (mathematical physics)|ensemble]] of the reference system with Hamiltonian <math>\mathcal{H}_0</math>. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as

where <math>S_0</math> is the [[entropy]], and <math>F</math> and <math>F_0</math> are [[Helmholtz free energy|Helmholtz free energies]]. The average is taken over the equilibrium [[Statistical ensemble (mathematical physics)|ensemble]] of the reference system with Hamiltonian <math>\mathcal{H}_0</math>. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as

where <math>S_0</math> is the entropy, and <math>F</math> and <math>F_0</math> are Helmholtz free energies. The average is taken over the equilibrium ensemble of the reference system with Hamiltonian <math>\mathcal{H}_0</math>. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as

其中，<math>S_0</math>是熵，而 <math>F</math>和<math>F_0</math>是亥姆霍兹自由能。平均值取参考系平衡系综的哈密顿量。在特殊情况下，参考哈密顿量是非相互作用系统的哈密顿量<math>\mathcal{H}_0</math>，因此可以写成

 

 

 

 

  <math>\mathcal{H}_0 = \sum_{i=1}^N h_i(\xi_i),</math>

  <math>\mathcal{H}_0 = \sum_{i=1}^N h_i(\xi_i),</math>

where <math>\xi_i</math> are the [[degrees of freedom (physics and chemistry)|degrees of freedom]] of the individual components of our statistical system (atoms, spins and so forth), one can consider sharpening the upper bound by minimizing the right side of the inequality. The minimizing reference system is then the "best" approximation to the true system using non-correlated degrees of freedom and is known as the '''mean-field approximation'''.

where <math>\xi_i</math> are the [[degrees of freedom (physics and chemistry)|degrees of freedom]] of the individual components of our statistical system (atoms, spins and so forth), one can consider sharpening the upper bound by minimizing the right side of the inequality. The minimizing reference system is then the "best" approximation to the true system using non-correlated degrees of freedom and is known as the '''mean-field approximation'''.

where <math>\xi_i</math> are the degrees of freedom of the individual components of our statistical system (atoms, spins and so forth), one can consider sharpening the upper bound by minimizing the right side of the inequality. The minimizing reference system is then the "best" approximation to the true system using non-correlated degrees of freedom and is known as the mean-field approximation.

<math>\xi_i</math> 是我们的统计系统的各个组成部分(原子、自旋等等)的自由度，我们可以考虑通过最小化不平等的右边来加强上限。最小化参考系是使用不相关自由度的真实系统的“最佳”近似，被称为平均场近似。

For the most common case that the target Hamiltonian contains only pairwise interactions, i.e.,

For the most common case that the target Hamiltonian contains only pairwise interactions, i.e.,

  <math>\mathcal{H} = \sum_{(i,j) \in \mathcal{P}} V_{i,j}(\xi_i, \xi_j),</math>

  <math>\mathcal{H} = \sum_{(i,j) \in \mathcal{P}} V_{i,j}(\xi_i, \xi_j),</math>

where <math>\mathcal{P}</math> is the set of pairs that interact, the minimizing procedure can be carried out formally. Define <math>\operatorname{Tr}_i f(\xi_i)</math> as the generalized sum of the observable <math>f</math> over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones). The approximating free energy is given by

where <math>\mathcal{P}</math> is the set of pairs that interact, the minimizing procedure can be carried out formally. Define <math>\operatorname{Tr}_i f(\xi_i)</math> as the generalized sum of the observable <math>f</math> over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones). The approximating free energy is given by

其中 < math > mathcal { p } </math > 是相互作用的对集合，最小化过程可以正式执行。定义{ Tr } _ i f (xi _ i) </math > 为单个分量自由度上可观测的 < math > f </math > 的广义和(离散变量的和，连续变量的积分)。给出了近似自由能的表达式

其中<math>\mathcal{P}</math>是相互作用的对集合，最小化过程可以正式执行。定义<math>\operatorname{Tr}_i f(\xi_i)</math>为单个分量自由度上可观测的<math>f</math>的广义和(离散变量的和，连续变量的积分)。给出了近似自由能的表达式

:<math>\begin{align}

:<math>\begin{align}

where <math>P^{(N)}_0(\xi_1, \xi_2, \dots, \xi_N)</math> is the probability to find the reference system in the state specified by the variables <math>(\xi_1, \xi_2, \dots, \xi_N)</math>. This probability is given by the normalized [[Boltzmann factor]]

where <math>P^{(N)}_0(\xi_1, \xi_2, \dots, \xi_N)</math> is the probability to find the reference system in the state specified by the variables <math>(\xi_1, \xi_2, \dots, \xi_N)</math>. This probability is given by the normalized [[Boltzmann factor]]

where <math>P^{(N)}_0(\xi_1, \xi_2, \dots, \xi_N)</math> is the probability to find the reference system in the state specified by the variables <math>(\xi_1, \xi_2, \dots, \xi_N)</math>. This probability is given by the normalized Boltzmann factor

其中<math>P^{(N)}_0(\xi_1, \xi_2, \dots, \xi_N)</math>是在变量<math>(\xi_1, \xi_2, \dots, \xi_N)</math>指定状态下找到参考系的概率。这个概率是由归一化玻尔兹曼因子给出的

 

其中 < math > p ^ {(n)} _ 0(xi _ 1，xi _ 2，dots，xi _ n) </math > 是在变量 < math > (xi _ 1，xi _ 2，dots，xi _ n) </math > 指定状态下找到参考系的概率。这个概率是由归一化玻尔兹曼因子给出的

: <math>\begin{align}

: <math>\begin{align}

where <math>Z_0</math> is the [[Partition function (statistical mechanics)|partition function]]. Thus

where <math>Z_0</math> is the [[Partition function (statistical mechanics)|partition function]]. Thus

where <math>Z_0</math> is the partition function. Thus

其中<math>Z_0</math>是配分函数。因此

 

:<math>\begin{align}

:<math>\begin{align}

In order to minimize, we take the derivative with respect to the single-degree-of-freedom probabilities <math>P^{(i)}_0</math> using a [[Lagrange multiplier]] to ensure proper normalization. The end result is the set of self-consistency equations

In order to minimize, we take the derivative with respect to the single-degree-of-freedom probabilities <math>P^{(i)}_0</math> using a [[Lagrange multiplier]] to ensure proper normalization. The end result is the set of self-consistency equations

In order to minimize, we take the derivative with respect to the single-degree-of-freedom probabilities <math>P^{(i)}_0</math> using a Lagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equations

为了得到最小化，我们对单自由度概率 <math>P^{(i)}_0</math> 求导，使用拉格朗日乘数来确保正确的归一化。最终得到的结果是自洽方程组

 

为了最小化，我们对单自由度概率 p ^ {(i)} _ 0 </math > 求导，使用拉格朗日乘数来确保正确的归一化。最终得到的结果是自洽方程组

 

  <math>P^{(i)}_0(\xi_i) = \frac{1}{Z_0} e^{-\beta h_i^{MF}(\xi_i)},\quad i = 1, 2, \ldots, N,</math>

  <math>P^{(i)}_0(\xi_i) = \frac{1}{Z_0} e^{-\beta h_i^{MF}(\xi_i)},\quad i = 1, 2, \ldots, N,</math>

where the mean field is given by

where the mean field is given by

 

 

 

==Applications==

==Applications 应用==

 

Mean-field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.

Mean-field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.

}}</ref>

}}</ref>

 

 

 

===Ising model===

===Ising model===

where we define <math>\delta s_i \equiv s_i - m_i</math>; this is the fluctuation of the spin.

where we define <math>\delta s_i \equiv s_i - m_i</math>; this is the fluctuation of the spin.

在这里我们定义“ math” ，这是自旋的涨落。

在这里我们定义<math>\delta s_i \equiv s_i - m_i</math>，这是自旋的涨落。

Consider the [[Ising model]] on a <math>d</math>-dimensional lattice. The Hamiltonian is given by

Consider the [[Ising model]] on a <math>d</math>-dimensional lattice. The Hamiltonian is given by

If we expand the right side, we obtain one term that is entirely dependent on the mean values of the spins and independent of the spin configurations. This is the trivial term, which does not affect the statistical properties of the system. The next term is the one involving the product of the mean value of the spin and the fluctuation value. Finally, the last term involves a product of two fluctuation values.

If we expand the right side, we obtain one term that is entirely dependent on the mean values of the spins and independent of the spin configurations. This is the trivial term, which does not affect the statistical properties of the system. The next term is the one involving the product of the mean value of the spin and the fluctuation value. Finally, the last term involves a product of two fluctuation values.