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--- 물리:bbgky_계층 [2022/04/19 13:37] – [푸아송 괄호 계산] jiwon
+++ 물리:bbgky_계층 [2022/04/21 17:24] – [푸아송 괄호 계산] jiwon
@@ Line 31: / Line 31: @@
 $$H' = \sum_{n=1}^s\sum_{i=s+1}^NV(\vec q_n-\vec q_j)$$
-여기에 리우빌 정리를 적용하면, $\rho_s$의 시간변화는
+여기에 [[물리:리우빌 정리]]를 적용하면, $\rho_s$의 시간변화는
 $$\frac{\partial\rho_s}{\partial t} = \int\prod_{i=s+1}^Nd^3p_id^3q_i\frac{\partial\rho}{\partial t} = -\int\prod_{i=s+1}^Nd^3p_id^3q_i \{\rho,H_s+H_{N-s}+H'\}$$
@@ Line 46: / Line 46: @@
 	\int \prod_{i=s+1}^N d^3p_id^3q_i\{\rho,H_{N-s}\} &= \int \prod_{i=s+1}^N d^3p_id^3q_i\sum_{j=1}^N\left[\frac{\partial\rho}{\partial\vec q_j}\cdot\frac{\partial H_{N-s}}{\partial\vec p_j}-\frac{\partial\rho}{\partial\vec p_j}\cdot\frac{\partial H_{N-s}}{\partial\vec q_j}\right]
 \end{align*}
-이고, $s+1\le j\le N$인 $j$에 대해
+이고, 각각의 $j$항을 따로 계산할 수 있다.
 \begin{align*}
-	\frac{\partial H_{N-s}}{\partial\vec p_j} &= \frac{\vec p_j}{m}\\
+	&\int d^3p_jd^3q_j\left[\frac{\partial\rho}{\partial\vec q_j}\cdot\frac{\partial H_{N-s}}{\partial\vec p_j}-\frac{\partial\rho}{\partial\vec p_j}\cdot\frac{\partial H_{N-s}}{\partial\vec q_j}\right]\\
-	\frac{\partial H_{N-s}}{\partial\vec q_j} &= \frac{\partial U(\vec q_j)}{\partial\vec q_j} + \frac12\sum_{k=s+1}\frac{V(\vec q_j-\vec q_k)}{\partial\vec q_j}
+	=&\int d^3p_j\left[\rho\frac{\partial H_{N-s}}{\partial\vec p_j}\right]_{\vec q_j\text{ at }\infty}-\int d^3q_j\left[\rho\frac{\partial H_{N-s}}{\partial\vec q_j}\right]_{\vec p_j\text{ at }\infty}+\int d^3p_jd^3q_j \rho\left[-\frac{\partial^2H_{N-s}}{\partial\vec p_j\partial\vec q_j}+\frac{\partial^2H_{N-s}}{\partial\vec q_j\partial\vec p_j}\right]\\
+	=&0
 \end{align*}
-이므로
+이는 우리가 고려하고자 하는 $s$개의 입자를 제외한 나머지들 끼리의 운동은 $\rho_s$에 아무 영향을 끼치지 않음을 의미한다.
+===세 번째 항===
+마찬가지로 푸아송 괄호를 풀어서 적어보면
 \begin{align*}
-	&= \int \prod_{i=s+1}^N d^3p_id^3q_i\sum_{j=s+1}^N\left[\frac{\partial\rho}{\partial\vec q_j}\cdot\frac{\vec p_j}{m}-\frac{\partial\rho}{\partial\vec p_j}\cdot\left(\frac{\partial U(\vec q_j)}{\partial\vec q_j} + \frac12\sum_{k=s+1}\frac{V(\vec q_j-\vec q_k)}{\partial\vec q_j}\right)\right]
+	\int \prod_{i=s+1}^N d^3p_id^3q_i\sum_{j=1}^N\left[\frac{\partial\rho}{\partial\vec q_j}\cdot\frac{\partial H'}{\partial\vec p_j}-\frac{\partial\rho}{\partial\vec p_j}\cdot\frac{\partial H'}{\partial\vec q_j}\right]
 \end{align*}
-이다. 위의 적분은 사실 $0$이 되는데 $j$하나를 골라서 적분해보면
+이다. $\partial H'/\partial\vec q_j=0$이 되고, $j\le s$일 때는
+\begin{align*}
+	\frac{\partial H'}{\partial\vec q_j} = \frac{\partial}{\partial\vec q_j}\left(\sum_{n=1}^s\sum_{i=s+1}^NV(\vec q_n-\vec q_i)\right) = \sum_{i=s+1}^NV(\vec q_j-\vec q_i)
+\end{align*}
+$s+1\le j\le N$일 때는
+\begin{align*}
+	\frac{\partial H'}{\partial\vec q_j} = \frac{\partial}{\partial\vec q_j}\left(\sum_{n=1}^s\sum_{i=s+1}^NV(\vec q_n-\vec q_i)\right) = \sum_{n=1}^sV(\vec q_n-\vec q_j)
+\end{align*}
+이므로, 이를 모두 모아서 적어보면
+\begin{align*}
+	&\int \prod_{i=s+1}^N d^3p_id^3q_i\sum_{j=1}^N\left[\frac{\partial\rho}{\partial\vec q_j}\cdot\frac{\partial H'}{\partial\vec p_j}-\frac{\partial\rho}{\partial\vec p_j}\cdot\frac{\partial H'}{\partial\vec q_j}\right]\\
+	=&\int \prod_{i=s+1}^N d^3p_id^3q_i\left[\sum_{n=1}^s\frac{\partial\rho}{\partial\vec p_n}\cdot\sum_{j=s+1}^N\frac{\partial V(\vec q_n-\vec q_j)}{\partial\vec q_n} + \sum_{j=s+1}^N\frac{\partial\rho}{\partial\vec p_j}\cdot\sum_{n=1}^s\frac{\partial V(\vec q_j-\vec q_n)}{\partial\vec  q_j}\right]
+\end{align*}
+이 된다.