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--- 물리:bbgky_계층 [2022/04/19 11:08] – [BBGKY 계층] jiwon
+++ 물리:bbgky_계층 [2022/04/21 16:47] – [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 38: / Line 38: @@
 ==== 푸아송 괄호 계산 ====
+===첫번째 항===
+$$\int \prod_{i=s+1}^N d^3p_id^3q_i \{\rho,H_s\} = \left\{\left(\int \prod_{i=s+1}^N d^3p_id^3q_i \rho\right),H_s\right\} = \{\rho_s,H_s\}$$
+===두번째 항===
+푸아송 괄호를 모두 풀어서 적어보면
+\begin{align*}
+	\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, 각각의 $j$항을 따로 계산할 수 있다.
+\begin{align*}
+	\frac{\partial H_{N-s}}{\partial\vec p_j} &= \frac{\vec p_j}{m}\\
+	\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}
+\end{align*}
+이므로
+\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]
+\end{align*}
+이다. 위의 적분은 사실 $0$이 되는데 $j$하나를 골라서 적분해보면