물리:포커-플랑크_방정식

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물리:포커-플랑크_방정식 [2026/03/10 12:50] – [포커-플랑크 방정식] admin물리:포커-플랑크_방정식 [2026/03/21 21:39] (current) – [같이 보기] admin
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 \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}. \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*} \end{eqnarray*}
 +사실 이 식은 꼭 이 형식론을 사용하지 않더라도 [[수학:편미분|편미분]]의 성질에 의해 바로 유도할 수 있는 매우 일반적인 관계식이다.
 좌변에 $x$를 곱하고 $x$에 대해 적분하면 $\delta(x-x_n)$의 적분에 의해 좌변에 $x$를 곱하고 $x$에 대해 적분하면 $\delta(x-x_n)$의 적분에 의해
 \begin{eqnarray*} \begin{eqnarray*}
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 \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). \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*} \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+따라서 특정한 시점 $\tau = m\Delta t$에서의 섭동의 변화 $\delta \xi_m$이 나중 시간 $t=n\Delta t$에서의 $\langle x \rangle_{\xi} - \langle x \rangle \equiv \delta \langle x \rangle$에 미치는 영향을 보면
 \begin{eqnarray*} \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)\\ \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)\\
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   * [[물리:경로적분_계산|경로적분]]   * [[물리:경로적분_계산|경로적분]]
   * [[수학:디락_델타_함수|디락 델타 함수]]   * [[수학:디락_델타_함수|디락 델타 함수]]
 +  * [[물리:구면_p-스핀_유리_모형|구면 $p$-스핀 유리 모형]]
 ======참고문헌====== ======참고문헌======
   * [[https://old.apctp.org/plan.php/statws2016|The 13th KIAS-APCTP Winter School on Statistical Physics]]   * [[https://old.apctp.org/plan.php/statws2016|The 13th KIAS-APCTP Winter School on Statistical Physics]]
   * [[http://events.kias.re.kr/h/winter2025|The 22nd KIAS-APCTP Winter School on Statistical Physics]]   * [[http://events.kias.re.kr/h/winter2025|The 22nd KIAS-APCTP Winter School on Statistical Physics]]
   * [[https://www.dsf.unica.it/~olla/statmec/|Piero Olla's lecture note]]   * [[https://www.dsf.unica.it/~olla/statmec/|Piero Olla's lecture note]]
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