물리:포커-플랑크_방정식

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물리:포커-플랑크_방정식 [2026/03/10 10:27] – [같이 보기] admin물리:포커-플랑크_방정식 [2026/03/10 12:50] (current) – [포커-플랑크 방정식] admin
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 \begin{eqnarray*} \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)\\ \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). &=& \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*} \end{eqnarray*}
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 $$\partial_t \rho(x,t) = -\partial_x \left[ a(x) \rho(x,t) \right] + \frac{1}{2} \partial_x^2 \rho(x,t).$$ $$\partial_t \rho(x,t) = -\partial_x \left[ a(x) \rho(x,t) \right] + \frac{1}{2} \partial_x^2 \rho(x,t).$$
 ======마틴-시지아-로즈(Martin-Siggia-Rose, MSR) 범함수 형식론====== ======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*}
  
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