Differences

This shows you the differences between two versions of the page.

--- 물리:포커-플랑크_방정식 [2026/03/10 11:30] – [마틴-시지아-로즈(Martin-Siggia-Rose, MSR) 범함수 형식론] admin
+++ 물리:포커-플랑크_방정식 [2026/03/10 12:50] (current) – [포커-플랑크 방정식] admin
@@ Line 16: / Line 16: @@
 \begin{eqnarray*}
 \rho(x, t+\Delta t) &=& \int dx' \int \frac{dk}{2\pi} \int \frac{d\Delta W}{\sqrt{2\pi \Delta t}} \exp\left\{ -ik \left[ x-x'-a(x') \Delta t-\Delta W \right] - \frac{(\Delta W)^2}{2\Delta t} \right\} \rho(x',t)\\
+&=& \int dx' \int \frac{dk}{2\pi} \exp\left\{ -ik \left[ x-x'-a(x') \Delta t \right] - \frac{1}{2} \Delta t k^2 \right\} \rho(x',t)\\
 &=& \frac{1}{\sqrt{2\pi \Delta t}} \int dx' \exp\left\{- \frac{[x-x'-a(x')\Delta t]^2}{2\Delta t} \right\} \rho(x',t).
 \end{eqnarray*}
@@ Line 71: / Line 72: @@
 \begin{eqnarray*}
 \rho_{\xi}(x,t) &\approx& \rho(x,t) + \sum_{j=0}^{n-1} \xi_j \left. \frac{\partial \rho_{\xi}(x,t)}{\partial \xi_j} \right|_{\forall \xi_i=0}.
+\end{eqnarray*}
+좌변에 $x$를 곱하고 $x$에 대해 적분하면 $\delta(x-x_n)$의 적분에 의해
+\begin{eqnarray*}
+\langle x \rangle_{\xi} = \int dx ~x~ \rho_{\xi}(x,t) &=& \int dx ~ x~ \mathcal{N} \int \left(\prod_{i=0}^{n-1} dx_i \right) \left(\prod_{i=0}^{n-1} dk_i \right) \left(\prod_{i=0}^{n-1} d\Delta W_i \right) \delta(x-x_n) \exp\left(\sum_{i=0}^{n-1} \left\{ -ik_{i+1} \left[ x_{i+1}-x_i-a(x_i) \Delta t - \Delta W_i -\xi_i \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \rho(x_0,0)\\
+&=& x_n ~\mathcal{N} \int \left(\prod_{i=0}^{n-1} dx_i \right) \left(\prod_{i=0}^{n-1} dk_i \right) \left(\prod_{i=0}^{n-1} d\Delta W_i \right) \exp\left(\sum_{i=0}^{n-1} \left\{ -ik_{i+1} \left[ x_{i+1}-x_i-a(x_i) \Delta t - \Delta W_i -\xi_i \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \rho(x_0,0).
+\end{eqnarray*}
+우변에 대해서도 마찬가지로 계산을 진행하면,
+\begin{eqnarray*}
+\langle x \rangle_{\xi} &\approx& \langle x \rangle + \sum_{j=0}^{n-1} \xi_j x_n ~\mathcal{N} \int \left(\prod_{i=0}^{n-1} dx_i \right) \left(\prod_{i=0}^{n-1} dk_i \right) \left(\prod_{i=0}^{n-1} d\Delta W_i \right) \left( ik_{j+1} \right) \exp\left(\sum_{i=0}^{n-1} \left\{ -ik_{i+1} \left[ x_{i+1}-x_i-a(x_i) \Delta t - \Delta W_i  \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \rho(x_0,0).
+\end{eqnarray*}
+따라서 특정한 시점 $\tau = m\Delta t$에서의 섭동의 변화 $\delta \xi_m$이 $\langle x \rangle_{\xi} - \langle x \rangle \equiv \delta \langle x \rangle$에 미치는 영향을 보면
+\begin{eqnarray*}
+\lim_{\forall \xi_i \to 0}\frac{\delta \langle x(t)\rangle}{\delta \xi(\tau)} &=& \mathcal{N} \int \left(\prod_{i=0}^{n-1} dx_i \right) \left(\prod_{i=0}^{n-1} dk_i \right) \left(\prod_{i=0}^{n-1} d\Delta W_i \right) \left( ik_{m+1} x_n \right) \exp\left(\sum_{i=0}^{n-1} \left\{ -ik_{i+1} \left[ x_{i+1}-x_i-a(x_i) \Delta t - \Delta W_i  \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \rho(x_0,0)\\
+&\approx& \langle ik(\tau) x(t) \rangle.
 \end{eqnarray*}