# 平均场理论

## Formal approach (Hamiltonian)

$\displaystyle{ \mathcal{H} = \mathcal{H}_0 + \Delta \mathcal{H} }$

$\displaystyle{ F \leq F_0 \ \stackrel{\mathrm{def}}{=}\ \langle \mathcal{H} \rangle_0 - T S_0, }$

$\displaystyle{ \mathcal{H}_0 = \sum_{i=1}^N h_i(\xi_i), }$

$\displaystyle{ \xi_i }$ 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。

$\displaystyle{ \mathcal{H} = \sum_{(i,j) \in \mathcal{P}} V_{i,j}(\xi_i, \xi_j), }$

\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} }

\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} }

\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} }

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

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

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

## Applications 应用

### 伊辛模型 Ising model

$\displaystyle{ \sum_{\langle i, j \rangle} }$表示所有最近邻居的和， $\displaystyle{ \langle i, j \rangle }$$\displaystyle{ s_i, s_j = \pm 1 }$ 是近邻伊辛旋数。

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

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

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



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

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


$\displaystyle{ z }$ 是协调数。

$\displaystyle{ h^\text{eff.} = h + J z m }$,它是外场 $\displaystyle{ h }$ 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 $\displaystyle{ d }$, $\displaystyle{ z = 2 d }$的超立方格)。

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

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

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

$\displaystyle{ h^\text{eff.} = h + J z m }$,它是外场 $\displaystyle{ h }$ 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 $\displaystyle{ d }$, $\displaystyle{ z = 2 d }$的超立方格)。

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

$\displaystyle{ T_\text{c} }$ 由以下关系给出：$\displaystyle{ T_\text{c} = \frac{J z}{k_B} }$.

• 当温度大于某个值时 $\displaystyle{ T_\text{c} }$,唯一解为 $\displaystyle{ m = 0 }$. 这个系统是顺磁性的。

• 对于 $\displaystyle{ T \lt T_\text{c} }$, 有两个非零解: $\displaystyle{ m = \pm m_0 }$. 这个系统是铁磁性的。

$\displaystyle{ T_\text{c} }$ 由以下关系给出： $\displaystyle{ T_\text{c} = \frac{J z}{k_B} }$.

### 在其他系统中的应用

• 研究金属-超导体的跃迁。在这种情况下，磁化的类似物是超导间隙
• 液晶中指导性场的拉普拉斯场为非零时产生的分子场。
• 在蛋白质结构预测中确定一个固定的蛋白质主链的最佳氨基酸侧链排列(见自洽平均场(生物学))。
• 测定复合材料的弹性性能。

## 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.

Category:Statistical mechanics

Category:Concepts in physics

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 $\displaystyle{ d }$-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 $\displaystyle{ m_i \equiv \langle s_i \rangle }$. 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 $\displaystyle{ \sum_{\langle i, j \rangle} }$ indicates summation over the pair of nearest neighbors $\displaystyle{ \langle i, j \rangle }$, and $\displaystyle{ s_i, s_j = \pm 1 }$ 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 $\displaystyle{ H = -J \sum_{\langle i, j \rangle} s_i s_j - h \sum_i s_i, }$ $\displaystyle{ H = -J \sum_{\langle i, j \rangle} s_i s_j - h \sum_i s_i, }$. ISBN 0-19-505316-8.