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--- 물리:포커-플랑크_방정식 [2026/03/07 21:24] – [포커-플랑크 방정식] admin
+++ 물리:포커-플랑크_방정식 [2026/03/10 12:50] (current) – [포커-플랑크 방정식] admin
@@ Line 2: / Line 2: @@
 간단한 예로서 다음과 같은 확률미분방정식
 $$dx = a[x(t)] dt + \sqrt{D} dW$$
-가 주어져 있다고 하자. $D$는 상수이고 $dW$는 위너 확률과정이다.
+가 주어져 있다고 하자. $D$는 상수로서 편의를 위해 $1$로 놓을 것이며, $dW$는 위너 확률과정이다.
 시간을 이산화할 때에 이토 해석을 따르면, $a$를 $t+\Delta t$가 아니라 $t$의 시점에서 계산하므로 다음처럼 적게 된다:
-$$x(t+\Delta t)-x(t) = a[x(t)] \Delta t + \sqrt{D} \Delta W.$$
+$$x(t+\Delta t)-x(t) = a[x(t)] \Delta t + \Delta W.$$
 시간 $t+\Delta t$에 위치 $x$에서 발견될 확률은 다음처럼 계산된다:
-$$\rho(x, t+\Delta t) = \int dx' \int d\Delta W P(\Delta W) \delta \left[ x-x'-a(x')\Delta t-\sqrt{D} \Delta W \right] \rho(x',t).$$
+$$\rho(x, t+\Delta t) = \int dx' \int d\Delta W P(\Delta W) \delta \left[ x-x'-a(x')\Delta t- \Delta W \right] \rho(x',t).$$
 즉 시간 $t$에 위치 $x'$에서 출발하되, $\Delta t$의 시간 동안 위의 이산 방정식을 통해 $x$에 도달할 수 있게끔 잡음이 발생할 확률들을 합하는 것이다.
 다음의 두 식을 대입한다:
 $$P(\Delta W) = \frac{1}{\sqrt{2\pi \Delta t}} \exp \left[ -\frac{(\Delta W)^2}{2\Delta t}\right]$$
-$$\delta(x) = \int \frac{dk}{2\pi} \exp[-ikx]$$.
+$$\delta(x) = \int \frac{dk}{2\pi} \exp[-ikx].$$
-그러면 그 결과는 아래와 같고
+그러면 그 결과는 아래와 같다:
 \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)\\
-&=& \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)
+&=& \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*}
 $y\equiv x-x'-a(x)\Delta t$라고 정의하자. 그러면 $x'-x = -y-a(x) \Delta t$이고, 지수 안의 분자에 들어가 있는 표현식은 다음처럼 적힌다:
 \begin{eqnarray*}
 x-x'-a(x') \Delta t &\approx& x-x' - \left[ a(x) + (x'-x) a'(x) \right] \Delta t\\
-&=& y-(x'-x)a'(x) \Delta t
+&=& y-(x'-x)a'(x) \Delta t\\
+&=& y+ \left[ y+a(x)\Delta t \right] a'(x) \Delta t.
 \end{eqnarray*}
-이때 $a'(x) \equiv da/dx$이다.
+이때 $a'(x) \equiv \frac{da}{dx}(x)$이다. 따라서 다음처럼 고쳐 적자:
+$$\frac{1}{2\Delta t}\left[ x-x'-a(x') \Delta t \right]^2 \approx \frac{y^2}{2\Delta t} + \left[y+a(x)\Delta t\right] a'(x)y$$
+\begin{eqnarray*}
+\rho(x',t) &\approx& \rho(x,t) + (x'-x) \rho'(x,t) + \frac{1}{2} (x'-x)^2 \rho''(x,t)\\
+&=& \rho(x,t) - \left[ y+a(x)\Delta t \right] \rho'(x,t) + \frac{1}{2} \left[ y+a(x)\Delta t \right]^2 \rho''(x,t).
+\end{eqnarray*}
+이때 $\rho'(x,t)\equiv \partial_x \rho(x,t)$이고 $\rho''(x,t) \equiv \partial_x^2 \rho(x,t)$이다.
+$x'$에서 $y$로 적분 변수를 바꿀 때에 자코비언이 $1$이므로 아래처럼 적을 수 있다:
+\begin{eqnarray*}
+\rho(x, t+\Delta t) &\approx& \frac{1}{\sqrt{2\pi \Delta t}} \int dy \exp\left(-\frac{y^2}{2\Delta t}\right) \exp\left\{ -\left[y+a(x)\Delta t\right] a'(x)y \right\} \rho(x',t)\\
+&\approx& \frac{1}{\sqrt{2\pi \Delta t}} \int dy \exp\left(-\frac{y^2}{2\Delta t}\right) \left\{ 1-\left[y+a(x)\Delta t\right] a'(x)y \right\} \rho(x',t)\\
+&\approx& \frac{1}{\sqrt{2\pi \Delta t}} \int dy \exp\left(-\frac{y^2}{2\Delta t}\right) \left\{ 1-\left[y+a(x)\Delta t\right] a'(x)y \right\} \left\{ \rho(x,t) - \left[ y+a(x)\Delta t \right] \rho'(x,t) + \frac{1}{2} \left[ y+a(x)\Delta t \right]^2 \rho''(x,t) \right\}.
+\end{eqnarray*}
+적분을 수행하고 $\Delta t$ 차수까지만 남겨놓자:
+\begin{eqnarray*}
+\rho(x, t+\Delta t) &\approx& \left[ 1-a'(x)\Delta t\right] \rho(x,t) -a(x) \Delta t\left[1-2a'(x) \Delta t \right] \rho'(x,t) + \frac{1}{2} \Delta t \left[ 1+a^2(x)\Delta t \right] \left[ 1-3a'(x)\Delta t\right] \rho''(x,t)\\
+&\approx& \left[ 1-a'(x)\Delta t \right] \rho(x,t) - a(x)\Delta t \rho'(x,t) + \frac{1}{2} \Delta t \rho''(x,t).
+\end{eqnarray*}
+$\Delta t\to 0$의 극한을 취하면 다음의 식을 얻는다:
+$$\partial_t \rho(x,t) = -\partial_x \left[ a(x) \rho(x,t) \right] + \frac{1}{2} \partial_x^2 \rho(x,t).$$
 ======마틴-시지아-로즈(Martin-Siggia-Rose, MSR) 범함수 형식론======
+시간 $t=0$에서의 초기 조건이 $x_0$일 때, 시간 $\Delta t$만큼의 진행을 $n$번 반복해서 $t=n\Delta t$까지 진행해보자.
+\begin{eqnarray*}
+\rho(x,t) &=&
+\mathcal{N} \int \left(\prod_{i=0}^{n-1} dx_i \right) \left(\prod_{i=0}^{n-1} dk_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 \right] - \frac{1}{2} \Delta t k_{i+1}^2 \right\} \right) \rho(x_0,0)\\
+&=& \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} 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 \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \rho(x_0,0).
+\end{eqnarray*}
+이때 $\mathcal{N}$은 계수들을 모아서 쓴 것이다. 외부로부터 아주 작은 섭동 $\xi_i$이 $\Delta W_i$에 더하여진다면 아래처럼 바뀔 것이다.
+\begin{eqnarray*}
+\rho_{\xi}(x,t) &=& \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)\\
+&=& \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 \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \exp \left(\sum_{j=0}^{n-1} ik_{j+1} \xi_j \right) \rho(x_0,0)\\
+&\approx& \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 \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \left(1+\sum_{j=0}^{n-1} ik_{j+1} \xi_j \right) \rho(x_0,0)\\
+&=& \rho(x,t) + \sum_{j=0}^{n-1} \left[ \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) \left( ik_{j+1} \xi_j \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) \right]\\
+&=& \rho(x,t) + \sum_{j=0}^{n-1} \xi_j \left[ \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) \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) \right].
+\end{eqnarray*}
+그런데 $\rho_{\xi}$를 $t=j\Delta t$로 표시되는 특정한 시점에서의 $\xi_j$로 미분하면 아래와 같으므로,
+\begin{eqnarray*}
+\frac{\partial \rho_{\xi}(x,t)}{\partial \xi_j} &=&\frac{\partial}{\partial \xi_j} \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)\\
+&=& \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) \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 - \xi_i \right] - \frac{\left( \Delta W_i \right)^2}{2\Delta t} \right\} \right) \rho(x_0,0)\\
+\left. \frac{\partial \rho_{\xi}(x,t)}{\partial \xi_j} \right|_{\forall \xi_i = 0} &=& \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) \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)\\
+&=& - \frac{\partial \rho(x,t)}{\partial x_{j+1}}
+\end{eqnarray*}
+두 식을 비교하면 다음을 알 수 있다:
+\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*}
 ======같이 보기======
-[[물리:경로적분_계산|경로적분]]
+  * [[물리:경로적분_계산|경로적분]]
+  * [[수학:디락_델타_함수|디락 델타 함수]]
 ======참고문헌======