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第169行： − F _ 0 & = 操作数名{ Tr }{1,2，ldots，n } cal { h }(xi _ 1，xi _ 2，ldots，xi _ n) p ^ {(n)} _ 0(xi _ 1，xi _ 2，ldots，xi _ n)+ 第195行：
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第311行： − 其中，[ math ] sum { langle i，j rangle } </math > 表示相邻的两个邻居 < math > langle i，j rangle </math > 和 < math > si，s _ j = pm 1 </math > 是相邻的伊辛自旋。+
Moved page from wikipedia:en:Mean-field theory (history)
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
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，s _ 0 </math > 是熵，而 < math > f </math > 和 < math > f _ 0 </math > 是亥姆霍兹自由能。用哈密顿数学方法求出参考系平衡系综的平均值。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
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
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.
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.
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。最小化的参考系是使用非相关自由度的真实系统的“最佳”近似，被称为平均场近似。
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。
<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>
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[数学]数学[数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学]
<math>\begin{align}
<math>\begin{align}
1.1.1.2.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3
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) \\
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),
<math>\begin{align}
<math>\begin{align}
1.1.1.2.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3
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)
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)} \\
<math>\begin{align}
<math>\begin{align}
1.1.1.2.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3
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) \\
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
为了最小化，我们对单自由度概率 p ^ {(i)} _ 0 </math > 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 <math>\sum_{\langle i, j \rangle}</math> indicates summation over the pair of nearest neighbors <math>\langle i, j \rangle</math>, and <math>s_i, s_j = \pm 1</math> are neighboring Ising spins.
where the <math>\sum_{\langle i, j \rangle}</math> indicates summation over the pair of nearest neighbors <math>\langle i, j \rangle</math>, and <math>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。
|isbn=0-19-505316-8
|isbn=0-19-505316-8