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--- 물리:tap_방정식 [2026/04/07 10:14] – [다른 스핀과의 관계] admin
+++ 물리:tap_방정식 [2026/04/08 11:37] (current) – [추가된 스핀] admin
@@ Line 254: / Line 254: @@
 결합 상수와 스핀 값들이 모두 독립적이므로 이 분포의 평균은 아래와 같다:
 $$\langle \tilde{h}_0 \rangle_{\backslash 0} = \sum_{i_2<\ldots<i_p} J_{0 i_2 \ldots i_p} \langle s_{i_2}\rangle_{\backslash 0} \cdots \langle s_{i_p} \rangle_{\backslash 0}.$$
+[[수학:인자_그래프|인자 그래프]]에서 계산한 방법대로 $\langle s_i \rangle_{\backslash 0}$을 $m_i \equiv \langle s_i \rangle$에 온사거 보정항(Onsager correction term)을 더함으로써 표현하면,
+\begin{eqnarray*}
+\langle \tilde{h}_0 \rangle_{\backslash 0} &\approx& \sum_{i_2<\ldots<i_p} J_{0 i_2 \ldots i_p} m_{i_2} \cdots m_{i_p} - \beta (1-q) m_0 \frac{p(p-1)}{2} q^{p-2}.
+\end{eqnarray*}
 분산은 다음처럼 계산되는데 이 중에서 대각항($i_1=k_1, \ldots, i_p=k_p$)만 살아남고 나머지 교차항들은 상쇄되어 평균적으로 0이라고 근사하면
 \begin{eqnarray*}
@@ Line 270: / Line 275: @@
 \begin{eqnarray*}
 P\left( s_0 \right) &\propto& \int d\tilde{h}_0 \exp\left[ -\frac{\beta r}{2}s_0^2 + \beta \tilde{h}_0 s_0 - \frac{\left(\tilde{h}_0 - \langle \tilde{h}_0 \rangle_{\backslash 0}\right)^2}{2V_0} \right]\\
-&=& \left(\langle \tilde{h}_0 \rangle_{\backslash 0} + \beta s_0 V_0 \right) \exp\left[ - \frac{\beta r}{2} s_0^2 + \beta \langle \tilde{h}_0 \rangle_{\backslash 0} s_0 + \frac12 \beta^2 V_0 \right],
+&\propto& \exp\left[ - \frac{\beta r}{2} s_0^2 + \beta \langle \tilde{h}_0 \rangle_{\backslash 0} s_0 + \frac12 \beta^2 s_0^2 V_0 \right],
 \end{eqnarray*}
 이로부터 $s_0$의 평균값과 (단, $r>\beta V_0$)
 \begin{eqnarray*}
-\langle s_0 \rangle &=& \frac{\int ds_0 s_0 P\left(s_0\right)}{\int ds_0 P\left(s_0\right)} = \frac{V_0}{\langle \tilde{h}_0 \rangle_{\backslash 0} r} + \frac{\langle \tilde{h}_0 \rangle_{\backslash 0}}{r - \beta V_0}.
+\langle s_0 \rangle &=& \frac{\int ds_0 s_0 P\left(s_0\right)}{\int ds_0 P\left(s_0\right)} = \frac{\langle \tilde{h}_0 \rangle_{\backslash 0}}{r - \beta V_0}.
 \end{eqnarray*}
 분산을 계산한다:
 \begin{eqnarray*}
-\langle \left( s_0 - \langle s_0 \rangle \right)^2 \rangle &=& \frac{\int ds_0 \left( s_0- \langle s_0 \rangle \right)^2 P\left(s_0\right)}{\int ds_0 P\left(s_0\right)} = - \left(\frac{V_0}{\langle \tilde{h}_0 \rangle_{\backslash 0} r}\right)^2 + \frac{1}{\beta \left(r - \beta V_0\right)}.
+\langle \left( s_0 - \langle s_0 \rangle \right)^2 \rangle &=& \frac{\int ds_0 \left( s_0- \langle s_0 \rangle \right)^2 P\left(s_0\right)}{\int ds_0 P\left(s_0\right)} = \frac{1}{\beta \left(r - \beta V_0\right)} = 1-q.
 \end{eqnarray*}
-구면 조건을 생각하면 이 양은 사실 $\langle s_0^2 \rangle - \langle s_0 \rangle^2 = 1-q$로도 쓸 수 있다.
+마지막의 표현식은 구면 조건과 $q \equiv N^{-1} \sum_i m_i^2$의 정의로부터 나온다. 이것을 이용하면 $s_0$가 실제로 추가되었을 때 그 평균값인 $m_0 \equiv \langle s_0 \rangle$에 대한 또다른 표현식은,
+\begin{eqnarray*}
+\frac{m_0}{\beta(1-q)} &=& \langle \tilde{h}_0 \rangle_{\backslash 0} \approx \sum_{i_2<\ldots<i_p} J_{0 i_2 \ldots i_p} m_{i_2} \cdots m_{i_p} - \beta (1-q) m_0 \frac{p(p-1)}{2} q^{p-2}.
+\end{eqnarray*}
+$\partial q/\partial m_i = 2m_i/N$에 유의하면, 위의 식은 아래와 같은 자유 에너지 밀도를 미분함으로써 얻을 수 있다:
+$$f = -\frac{1}{N} \sum_{i_1<\ldots<i_p} J_{i_1 \ldots i_p} m_{i_1} \cdots m_{i_p} - \frac{1}{2\beta} \ln(1-q) - \frac{\beta}{4} \left[ (p-1) q^p - pq^{p-1} +1 \right].$$
+/*
 =====다른 스핀과의 관계=====
 스핀 $0$가 아닌 다른 스핀 $i$를 포함하여 식을 적어보자.
@@ Line 293: / Line 305: @@
 \begin{eqnarray*}
 \Bigl< \left( \tilde{h}_i - \langle \tilde{h}_i \rangle_{\backslash 0} \right)^2 \Bigr>_{\backslash 0} &=& \langle \tilde{h}_i^2 \rangle_{\backslash 0} - \langle \tilde{h}_i \rangle_{\backslash 0}^2\\
-&\approx& \frac{1}{\left[(p-2)!\right]^2} (p-2)! \sum_{i_3,\ldots,i_p} J_{0 i i_3 \ldots i_p}^2 \left( \langle s_{i_3}^2 \rangle \cdots \langle s_{i_p}^2 \rangle - \langle s_{i_3} \rangle^2 \cdots \langle s_{i_p}\rangle^2 \right)\\
+&\approx& \frac{1}{\left[(p-2)!\right]^2} (p-2)! \sum_{i_3,\ldots,i_p} J_{0 i i_3 \ldots i_p}^2 \left( \langle s_{i_3}^2 \rangle_{\backslash 0} \cdots \langle s_{i_p}^2 \rangle_{\backslash 0} - \langle s_{i_3} \rangle_{\backslash 0}^2 \cdots \langle s_{i_p}\rangle_{\backslash 0}^2 \right)\\
 &\approx& \frac{1}{(p-2)!} \sum_{i_3,\ldots,i_p} \frac{p!}{2N^{p-1}} \left( 1 - q^{p-2} \right)\\
 &=& \frac{1}{(p-2)!} N^{p-2} \frac{p!}{2N^{p-1}} \left( 1 - q^{p-2} \right)\\
@@ Line 306: / Line 318: @@
 P\left(s_0, s_i, \tilde{h}_i, \tilde{h}_i' \right) &\propto& \exp\left[ -\frac{\beta r}{2} s_0^2 + \beta s_0 \left(s_i \tilde{h}_i + \tilde{h}_i' \right) - \frac{\left( s_i - \langle s_i \rangle_{\backslash 0} \right)^2}{2(1-q)} - N \frac{\left( \tilde{h}_i - \langle \tilde{h}_i \rangle_{\backslash 0} \right)^2}{p(p-1)\left( 1-q^{p-1} \right)} - \frac{\left( \tilde{h}_i' - \langle \tilde{h}_i' \rangle_{\backslash 0} \right)^2}{p\left( 1-q^{p-1} \right)} \right].
 \end{eqnarray*}
+주변화를 거치면 아래와 같은 적분 결과를 얻고
+\begin{eqnarray*}
+P\left(s_0, s_i \right) &=& \int d\tilde{h}_i d\tilde{h}_i' P\left(s_0, s_i, \tilde{h}_i, \tilde{h}_i' \right)\\
+&\propto& \exp\left[ -\frac{\beta r}{2}s_0^2+\frac{p\beta^2}{4}\left(1-q^{p-1}\right) s_0^2 + \beta s_0 \left( s_i \langle \tilde{h}_i \rangle_{\backslash 0} + \langle \tilde{h}_i' \rangle_{\backslash 0} \right)-\frac{\left(s_i-\langle s_i \rangle_{\backslash 0}\right)^2}{2(1-q)} -\frac{1}{4N}\beta^2 (1-p)p \left(1-q^{p-1}\right) s_0^2 s_i^2 \right]\\
+&\approx& \exp\left[ -\frac{s_0^2}{2(1-q)} + \beta s_0 \left( s_i \langle \tilde{h}_i \rangle_{\backslash 0} + \langle \tilde{h}_i' \rangle_{\backslash 0} \right)-\frac{\left(s_i-\langle s_i \rangle_{\backslash 0}\right)^2}{2(1-q)} \right],
+\end{eqnarray*}
+지수 중 $N^{-1}$에 비례하는 항은 $N\to\infty$에서 무시했다. 이제 주변화로 $P(s_i) = \int ds_0 P(s_0, s_i)$를 얻은 다음 $\langle s_i \rangle$을 $\tilde{h}_i \sim O\left(N^{-1/2}\right)$의 일차항까지 계산해보면,
+\begin{eqnarray*}
+\langle s_i \rangle &=& \frac{\int ds_i ~s_i P\left( s_i \right)}{\int ds_i P\left( s_i \right)} = \frac{\langle s_i \rangle_{\backslash 0} + \beta^2 (1-q)^2 \langle \tilde{h}_i \rangle_{\backslash 0} \langle \tilde{h}_i' \rangle_{\backslash 0}}{1-\beta^2(1-q)^2 \langle \tilde{h}_i \rangle_{\backslash 0}^2} \approx \langle s_i \rangle_{\backslash 0} + \beta^2 (1-q)^2 \langle \tilde{h}_i \rangle_{\backslash 0} \langle \tilde{h}_i' \rangle_{\backslash 0}\\
+&=& \langle s_i \rangle_{\backslash 0} + \beta (1-q) \langle s_0 \rangle \langle \tilde{h}_i' \rangle_{\backslash 0}\\
+&\approx& \langle s_i \rangle_{\backslash 0} + \beta (1-q) \langle s_0 \rangle \sum_{i_3<\ldots<i_p} J_{0 i i_3 \ldots i_p} \langle s_{i_3} \rangle_{\backslash 0} \cdots \langle s_{i_p} \rangle_{\backslash 0}.
+\end{eqnarray*}
+*/
 ======함께 보기======
   * [[신경망:공동_방법|공동 방법]]