平均场理论

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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.[1] 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. 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[2], artificial intelligence, epidemic models,[3] queueing theory,[4] computer-network performance and game theory,[5] as in the quantal response equilibrium.

MFT has since been applied to a wide range of fields outside of physics, including statistical inference, graphical models, neuroscience, artificial intelligence, epidemic models, queueing theory, computer-network performance and game theory, as in the quantal response equilibrium.

此后,MFT 被广泛应用于物理学以外的领域,包括推论统计学、图形模型、神经科学、人工智能、传染病模型、排队论、计算机网络性能和博弈论,如量子反应均衡。


Origins

The ideas first appeared in physics (statistical mechanics) in the work of Pierre Curie[6] and Pierre Weiss to describe phase transitions.[7] 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(统计力学)的 Pierre Curie 和 Pierre Weiss 描述相变的著作中。MFT 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。


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.

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(有时是无限)自由度的系统通常难以精确地求解或以封闭的解析形式计算,除了一些简单的情况(例如:。高斯随机场理论,一维伊辛模型)。组合问题经常出现,使得计算一个系统的配分函数/值变得困难。MFT 是一种近似方法,它常常使原问题变得可解,易于计算。有时,MFT 给出非常精确的近似值。


In 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 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".

在场论中,哈密顿量可以根据场平均周围起伏的大小来展开。在这种背景下,MFT 可以看作是哈密顿量在涨落中的“零阶”展开。物理上,这意味着 MFT 系统没有波动,但这与“平均场”取代所有相互作用的观点不谋而合。


Quite often, MFT provides a convenient launch point to studying higher-order fluctuations. For example, when computing the partition function, studying the combinatorics of the interaction terms in the Hamiltonian can sometimes at best produce perturbative results or Feynman diagrams 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, studying the combinatorics of the interaction terms in the Hamiltonian can sometimes at best produce perturbative results or Feynman diagrams that correct the mean-field approximation.

MFT 常常为研究高阶波动提供了一个方便的起点。例如,当计算配分函数时,研究哈密顿量中相互作用项的组合有时候最多只能产生微扰结果或修正平均场近似的费曼图。


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.

一般来说,维数在确定平均场方法是否适用于任何特定问题时起着重要作用。有时存在一个临界维度,高于该维度 MFT 有效,低于该维度 MFT 无效。


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.

启发式地,MFT 中的许多交互被一个有效的交互所取代。因此,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 MFT 更加精确。这在高维情况下是正确的,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)。金兹堡准则是波动如何使 MFT 成为一个糟糕的近似的正式表达,通常取决于感兴趣的系统中的空间维数。


Formal approach (Hamiltonian)

The formal basis for mean-field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian

The formal basis for mean-field theory is the Bogoliubov inequality. This inequality states that the 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


[math]\displaystyle{ \mathcal{H} = \mathcal{H}_0 + \Delta \mathcal{H} }[/math]
[math]\displaystyle{ \mathcal{H} = \mathcal{H}_0 + \Delta \mathcal{H} }[/math]

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has the following upper bound:

has the following upper bound:

有下面的上界:


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

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where [math]\displaystyle{ S_0 }[/math] is the entropy, and [math]\displaystyle{ F }[/math] and [math]\displaystyle{ F_0 }[/math] are Helmholtz free energies. The average is taken over the equilibrium ensemble of the reference system with Hamiltonian [math]\displaystyle{ \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]\displaystyle{ S_0 }[/math] is the entropy, and [math]\displaystyle{ F }[/math] and [math]\displaystyle{ F_0 }[/math] are Helmholtz free energies. The average is taken over the equilibrium ensemble of the reference system with Hamiltonian [math]\displaystyle{ \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,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,因此可以写成


[math]\displaystyle{ \mathcal{H}_0 = \sum_{i=1}^N h_i(\xi_i), }[/math]
[math]\displaystyle{ \mathcal{H}_0 = \sum_{i=1}^N h_i(\xi_i), }[/math]

0 = sum { i = 1} ^ n h _ i (xi _ i) ,</math >


where [math]\displaystyle{ \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.

where [math]\displaystyle{ \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.

如果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.,

对于最常见的情况,目标哈密顿函数只包含成对相互作用,例如,


[math]\displaystyle{ \mathcal{H} = \sum_{(i,j) \in \mathcal{P}} V_{i,j}(\xi_i, \xi_j), }[/math]
[math]\displaystyle{ \mathcal{H} = \sum_{(i,j) \in \mathcal{P}} V_{i,j}(\xi_i, \xi_j), }[/math]

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where [math]\displaystyle{ \mathcal{P} }[/math] is the set of pairs that interact, the minimizing procedure can be carried out formally. Define [math]\displaystyle{ \operatorname{Tr}_i f(\xi_i) }[/math] as the generalized sum of the observable [math]\displaystyle{ 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]\displaystyle{ \mathcal{P} }[/math] is the set of pairs that interact, the minimizing procedure can be carried out formally. Define [math]\displaystyle{ \operatorname{Tr}_i f(\xi_i) }[/math] as the generalized sum of the observable [math]\displaystyle{ 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

[math]\displaystyle{ \begin{align} \lt math\gt \begin{align} 1.1.1.2.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3 F_0 &= \operatorname{Tr}_{1,2,\ldots,N} \mathcal{H}(\xi_1, \xi_2, \ldots, \xi_N) P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) \\ F_0 &= \operatorname{Tr}_{1,2,\ldots,N} \mathcal{H}(\xi_1, \xi_2, \ldots, \xi_N) P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) \\ 1,2,ldots,n } cal { h }(xi _ 1,xi _ 2,ldots,xi _ n) p ^ {(n)} _ 0(xi _ 1,xi _ 2,ldots,xi _ n) &+ kT \,\operatorname{Tr}_{1,2,\ldots,N} P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) \log P^{(N)}_0(\xi_1, \xi_2, \ldots,\xi_N), &+ kT \,\operatorname{Tr}_{1,2,\ldots,N} P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) \log P^{(N)}_0(\xi_1, \xi_2, \ldots,\xi_N), 1,2,ldots,n } p ^ {(n)} _ 0(xi _ 1,xi _ 2,ldots,xi _ n) log p ^ {(n)} _ 0(xi _ 1,xi _ 2,ldots,xi _ n) \end{align} }[/math]

\end{align}</math>

结束{ align } </math >


where [math]\displaystyle{ 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]\displaystyle{ (\xi_1, \xi_2, \dots, \xi_N) }[/math]. This probability is given by the normalized Boltzmann factor

where [math]\displaystyle{ 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]\displaystyle{ (\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

[math]\displaystyle{ \begin{align} \lt math\gt \begin{align} 1.1.1.2.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3 P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) P ^ {(n)} _ 0(xi _ 1,xi _ 2,ldots,xi _ n) &= \frac{1}{Z^{(N)}_0} e^{-\beta \mathcal{H}_0(\xi_1, \xi_2, \ldots, \xi_N)} \\ &= \frac{1}{Z^{(N)}_0} e^{-\beta \mathcal{H}_0(\xi_1, \xi_2, \ldots, \xi_N)} \\ 0(xi _ 1,xi _ 2,ldots,xi _ n)} &= \prod_{i=1}^N \frac{1}{Z_0} e^{-\beta h_i(\xi_i)} \ \stackrel{\mathrm{def}}{=}\ \prod_{i=1}^N P^{(i)}_0(\xi_i), &= \prod_{i=1}^N \frac{1}{Z_0} e^{-\beta h_i(\xi_i)} \ \stackrel{\mathrm{def}}{=}\ \prod_{i=1}^N P^{(i)}_0(\xi_i), 1} ^ n frac {1}{ z _ 0} e ^ {-beta h _ i (xi _ i)} stackrel { mathrm { def }{ = } prod _ { i = 1} ^ n p ^ {(i)} _ 0(xi _ i) , \end{align} }[/math]

\end{align}</math>

结束{ align } </math >


where [math]\displaystyle{ Z_0 }[/math] is the partition function. Thus

where [math]\displaystyle{ Z_0 }[/math] is the partition function. Thus

其中 z 0是配分函数。因此

[math]\displaystyle{ \begin{align} \lt math\gt \begin{align} 1.1.1.2.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3 F_0 &= \sum_{(i,j) \in \mathcal{P}} \operatorname{Tr}_{i,j} V_{i,j}(\xi_i, \xi_j) P^{(i)}_0(\xi_i) P^{(j)}_0(\xi_j) \\ F_0 &= \sum_{(i,j) \in \mathcal{P}} \operatorname{Tr}_{i,j} V_{i,j}(\xi_i, \xi_j) P^{(i)}_0(\xi_i) P^{(j)}_0(\xi_j) \\ F _ 0 & = sum _ {(i,j) in mathcal { p }算子名{ Tr } _ { i,j } v _ { i,j }(xi _ i,xi _ j) p ^ {(i)} _ 0(xi _ i) p ^ {(j)} _ 0(xi _ j)) &+ kT \sum_{i=1}^N \operatorname{Tr}_i P^{(i)}_0(\xi_i) \log P^{(i)}_0(\xi_i). &+ kT \sum_{i=1}^N \operatorname{Tr}_i P^{(i)}_0(\xi_i) \log P^{(i)}_0(\xi_i). & + kT sum { i = 1} ^ n 操作符名称{ Tr } i p ^ {(i)} _ 0(xi _ i) log p ^ {(i)} _ 0(xi _ i)。 \end{align} }[/math]

\end{align}</math>

结束{ align } </math >


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

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

平均场是从哪里来的

[math]\displaystyle{ h_i^\text{MF}(\xi_i) = \sum_{\{j \mid (i,j) \in \mathcal{P}\}} \operatorname{Tr}_j V_{i,j}(\xi_i, \xi_j) P^{(j)}_0(\xi_j). }[/math]
[math]\displaystyle{ h_i^\text{MF}(\xi_i) = \sum_{\{j \mid (i,j) \in \mathcal{P}\}} \operatorname{Tr}_j V_{i,j}(\xi_i, \xi_j) P^{(j)}_0(\xi_j). }[/math]

数学运算符名称{ Tr } j v { i,j }(xi _ i,xi _ j) p ^ { j } _ 0(xi _ j)


Applications

Mean-field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.[8]

[math]\displaystyle{ H = -J \sum_{\langle i, j \rangle} (m_i + \delta s_i) (m_j + \delta s_j) - h \sum_i s_i, }[/math]

< math > h =-j sum { langle i,j rangle }(m _ i + delta s _ i)(m _ j + delta s _ j)-h sum i s _ i,</math >


Ising model

where we define [math]\displaystyle{ \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。

Consider the Ising model on a [math]\displaystyle{ d }[/math]-dimensional lattice. The Hamiltonian is given by

[math]\displaystyle{ H = -J \sum_{\langle i, j \rangle} s_i s_j - h \sum_i s_i, }[/math]

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.

如果我们展开右边,我们得到一个项,它完全依赖于自旋的平均值,与自旋构型无关。这是一个平凡的术语,它不影响系统的统计特性。下一项是自旋平均值与涨落值的乘积。最后,最后一项涉及两个涨落值的乘积。

where the [math]\displaystyle{ \sum_{\langle i, j \rangle} }[/math] indicates summation over the pair of nearest neighbors [math]\displaystyle{ \langle i, j \rangle }[/math], and [math]\displaystyle{ s_i, s_j = \pm 1 }[/math] are neighboring Ising spins.


The mean-field approximation consists of neglecting this second-order fluctuation term:

平均场近似忽略了这个二阶涨落项:

Let us transform our spin variable by introducing the fluctuation from its mean value [math]\displaystyle{ m_i \equiv \langle s_i \rangle }[/math]. We may rewrite the Hamiltonian as

[math]\displaystyle{ H \approx H^\text{MF} \equiv -J \sum_{\langle i, j \rangle} (m_i m_j + m_i \delta s_j + m_j \delta s_i) - h \sum_i s_i. }[/math]

(m _ i m _ j + m _ i delta s _ j + m _ j delta s _ i)-h sum _ i s _ i.数学

[math]\displaystyle{ H = -J \sum_{\langle i, j \rangle} (m_i + \delta s_i) (m_j + \delta s_j) - h \sum_i s_i, }[/math]


These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

这些涨落在低维度上增强,使 MFT 成为高维度的更好近似值。

where we define [math]\displaystyle{ \delta s_i \equiv s_i - m_i }[/math]; this is the fluctuation of the spin.


Again, the summand can be reexpanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields

同样,可以重新扩大这一需求。另外,由于伊辛链具有平移不变性,我们希望每个自旋的平均值与位置无关。这就产生了

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.


[math]\displaystyle{ H^\text{MF} = -J \sum_{\langle i, j \rangle} \big(m^2 + 2m(s_i - m)\big) - h \sum_i s_i. }[/math]

=-j sum { langle i,j rangle } big (m ^ 2 + 2m (s _ i-m) big)-h sum _ i s _ i. </math >

The mean-field approximation consists of neglecting this second-order fluctuation term:

[math]\displaystyle{ H \approx H^\text{MF} \equiv -J \sum_{\langle i, j \rangle} (m_i m_j + m_i \delta s_j + m_j \delta s_i) - h \sum_i s_i. }[/math]

The summation over neighboring spins can be rewritten as [math]\displaystyle{ \sum_{\langle i, j \rangle} = \frac{1}{2} \sum_i \sum_{j \in nn(i)} }[/math], where [math]\displaystyle{ nn(i) }[/math] means "nearest neighbor of [math]\displaystyle{ i }[/math]", and the [math]\displaystyle{ 1/2 }[/math] prefactor avoids double counting, since each bond participates in two spins. Simplifying leads to the final expression

相邻自旋的和可以重写为 < math > sum _ { langle i,j rangle } = frac {1}{2} sum _ i sum _ { j in nn (i)} </math > ,其中 < math > nn (i) </math > 表示“ < math > i </math > 的最近邻” ,< math > 1/2 </math > 前因子避免了重复计算,因为每个键参与两个自旋。简化导致最终的表达式


These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

[math]\displaystyle{ H^\text{MF} = \frac{J m^2 N z}{2} - \underbrace{(h + m J z)}_{h^\text{eff.}} \sum_i s_i, }[/math]

2}-underbrace {(h + m j z)} _ h ^ text { eff. }数学,数学


Again, the summand can be reexpanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields

where [math]\displaystyle{ z }[/math] is the coordination number. At this point, the Ising Hamiltonian has been decoupled into a sum of one-body Hamiltonians with an effective mean field [math]\displaystyle{ h^\text{eff.} = h + J z m }[/math], which is the sum of the external field [math]\displaystyle{ h }[/math] and of the mean field induced by the neighboring spins. It is worth noting that this mean field directly depends on the number of nearest neighbors and thus on the dimension of the system (for instance, for a hypercubic lattice of dimension [math]\displaystyle{ d }[/math], [math]\displaystyle{ z = 2 d }[/math]).

其中 z </math > 是协调数。在这一点上,伊辛哈密顿函数已经解耦为一个有效平均场 < math > h ^ text { eff. }的单体哈密顿函数之和= h + j z m </math > ,它是外场和相邻自旋引起的平均场的总和。值得注意的是,这个平均值域直接取决于最近邻居的数量,因此取决于系统的维数(例如,对于维数为 < math > d </math > ,< math > z = 2 d </math > 的超立方格)。


[math]\displaystyle{ H^\text{MF} = -J \sum_{\langle i, j \rangle} \big(m^2 + 2m(s_i - m)\big) - h \sum_i s_i. }[/math]

Substituting this Hamiltonian into the partition function and solving the effective 1D problem, we obtain

把这个哈密顿量代入配分函数,求解有效的一维问题,我们得到了


The summation over neighboring spins can be rewritten as [math]\displaystyle{ \sum_{\langle i, j \rangle} = \frac{1}{2} \sum_i \sum_{j \in nn(i)} }[/math], where [math]\displaystyle{ nn(i) }[/math] means "nearest neighbor of [math]\displaystyle{ i }[/math]", and the [math]\displaystyle{ 1/2 }[/math] prefactor avoids double counting, since each bond participates in two spins. Simplifying leads to the final expression

[math]\displaystyle{  Z = e^{-\frac{\beta J m^2 Nz}{2}} \left[2 \cosh\left(\frac{h + m J z}{k_\text{B} T}\right)\right]^N, }[/math]

左[2 cosh left (frac { h + m j }{ k _ text { b } t } right)] ^ n,</math >


[math]\displaystyle{ H^\text{MF} = \frac{J m^2 N z}{2} - \underbrace{(h + m J z)}_{h^\text{eff.}} \sum_i s_i, }[/math]

where [math]\displaystyle{ N }[/math] is the number of lattice sites. This is a closed and exact expression for the partition function of the system. We may obtain the free energy of the system and calculate critical exponents. In particular, we can obtain the magnetization [math]\displaystyle{ m }[/math] as a function of [math]\displaystyle{ h^\text{eff.} }[/math].

其中 < math > n </math > 是格点的数量。这是一个封闭而精确的系统配分函数表达式。我们可以得到系统的自由能和计算临界指数。特别地,我们可以得到磁化作为 < math > h ^ text { eff 的函数。{/math > .


where [math]\displaystyle{ z }[/math] is the coordination number. At this point, the Ising Hamiltonian has been decoupled into a sum of one-body Hamiltonians with an effective mean field [math]\displaystyle{ h^\text{eff.} = h + J z m }[/math], which is the sum of the external field [math]\displaystyle{ h }[/math] and of the mean field induced by the neighboring spins. It is worth noting that this mean field directly depends on the number of nearest neighbors and thus on the dimension of the system (for instance, for a hypercubic lattice of dimension [math]\displaystyle{ d }[/math], [math]\displaystyle{ z = 2 d }[/math]).

We thus have two equations between [math]\displaystyle{ m }[/math] and [math]\displaystyle{ h^\text{eff.} }[/math], allowing us to determine [math]\displaystyle{ m }[/math] as a function of temperature. This leads to the following observation:

因此,我们有两个方程在 < math > 和 < math > h ^ text { eff 之间。这样我们就可以确定温度的函数。这导致了以下结论:


Substituting this Hamiltonian into the partition function and solving the effective 1D problem, we obtain


[math]\displaystyle{ Z = e^{-\frac{\beta J m^2 Nz}{2}} \left[2 \cosh\left(\frac{h + m J z}{k_\text{B} T}\right)\right]^N, }[/math]

[math]\displaystyle{ T_\text{c} }[/math] is given by the following relation: [math]\displaystyle{ T_\text{c} = \frac{J z}{k_B} }[/math].

“数学”是通过以下关系给出的: < math > t _ text { c } = frac { j }{ k _ b } </math > 。


where [math]\displaystyle{ N }[/math] is the number of lattice sites. This is a closed and exact expression for the partition function of the system. We may obtain the free energy of the system and calculate critical exponents. In particular, we can obtain the magnetization [math]\displaystyle{ m }[/math] as a function of [math]\displaystyle{ h^\text{eff.} }[/math].

This shows that MFT can account for the ferromagnetic phase transition.

这说明 MFT 可以解释铁磁相变。


We thus have two equations between [math]\displaystyle{ m }[/math] and [math]\displaystyle{ h^\text{eff.} }[/math], allowing us to determine [math]\displaystyle{ m }[/math] as a function of temperature. This leads to the following observation:

  • For temperatures greater than a certain value [math]\displaystyle{ T_\text{c} }[/math], the only solution is [math]\displaystyle{ m = 0 }[/math]. The system is paramagnetic.

Similarly, MFT can be applied to other types of Hamiltonian as in the following cases:

同样,MFT 也可以应用于其他类型的哈密顿量,如下列情况:

  • For [math]\displaystyle{ T \lt T_\text{c} }[/math], there are two non-zero solutions: [math]\displaystyle{ m = \pm m_0 }[/math]. The system is ferromagnetic.


[math]\displaystyle{ T_\text{c} }[/math] is given by the following relation: [math]\displaystyle{ T_\text{c} = \frac{J z}{k_B} }[/math].


This shows that MFT can account for the ferromagnetic phase transition.


Application to other systems

Similarly, MFT can be applied to other types of Hamiltonian as in the following cases:

  • To study the metal–superconductor transition. In this case, the analog of the magnetization is the superconducting gap [math]\displaystyle{ \Delta }[/math].

In mean-field theory, the mean field appearing in the single-site problem is a scalar or vectorial time-independent quantity. However, this need not always be the case: in a variant of mean-field theory called dynamical mean-field theory (DMFT), the mean field becomes a time-dependent quantity. For instance, DMFT can be applied to the Hubbard model to study the metal–Mott-insulator transition.

在平均场理论中,单点问题中出现的平均场是一个与时间无关的标量或向量。然而,情况并非总是如此: 在一种称为动态平均场理论(DMFT)的平均场理论的变体中,平均场变成了一个与时间有关的量。例如,DMFT 可以应用于哈伯德模型来研究金属-莫特-绝缘体的转变。


Extension to time-dependent mean fields


In mean-field theory, the mean field appearing in the single-site problem is a scalar or vectorial time-independent quantity. However, this need not always be the case: in a variant of mean-field theory called dynamical mean-field theory (DMFT), the mean field becomes a time-dependent quantity. For instance, DMFT can be applied to the Hubbard model to study the metal–Mott-insulator transition.


See also

Category:Statistical mechanics

类别: 统计力学

Category:Concepts in physics

分类: 物理概念


This page was moved from wikipedia:en:Mean-field theory. Its edit history can be viewed at 平均场理论/edithistory

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  8. Mean-field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions. 平均场理论可以应用于许多物理系统,以便研究相变等现象。 Stanley Consider the Ising model on a [math]\displaystyle{ d }[/math]-dimensional lattice. The Hamiltonian is given by 考虑一个 < math > d </math > 维格上的 Ising 模型。哈密顿函数是由, H. E. (1971 Let us transform our spin variable by introducing the fluctuation from its mean value [math]\displaystyle{ m_i \equiv \langle s_i \rangle }[/math]. We may rewrite the Hamiltonian as 让我们通过引入涨落来转换自旋变量,从它的平均值 < math > m _ i = l _ i rangle </math > 。我们可以把哈密顿函数改写成). "Mean Field Theory of Magnetic Phase Transitions where the [math]\displaystyle{ \sum_{\langle i, j \rangle} }[/math] indicates summation over the pair of nearest neighbors [math]\displaystyle{ \langle i, j \rangle }[/math], and [math]\displaystyle{ s_i, s_j = \pm 1 }[/math] are neighboring Ising spins. 其中 < math > sum { langle i,j rangle } </math > 表示对最近邻居 < math > langle i,j rangle </math > ,和 < math > s i,s j = pm 1 </math > 是邻近的 Ising 自旋。". Introduction to Phase Transitions and Critical Phenomena. Oxford University Press [math]\displaystyle{ H = -J \sum_{\langle i, j \rangle} s_i s_j - h \sum_i s_i, }[/math] [math]\displaystyle{ H = -J \sum_{\langle i, j \rangle} s_i s_j - h \sum_i s_i, }[/math]. ISBN 0-19-505316-8.