# 福克-普朗克方程

（重定向自佛客－普朗克方程式

${\displaystyle {\frac {\partial }{\partial t}}f(x,t)=-{\frac {\partial }{\partial x}}\left[D_{1}(x,t)f(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D_{2}(x,t)f(x,t)\right].}$

${\displaystyle N}$ 維空間中的福克-普朗克方程是

${\displaystyle {\frac {\partial f}{\partial t}}=-\sum _{i=1}^{N}{\frac {\partial }{\partial x_{i}}}\left[D_{i}^{1}(x_{1},\ldots ,x_{N})f\right]+\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}\left[D_{ij}^{2}(x_{1},\ldots ,x_{N})f\right],}$ ${\displaystyle x_{i}}$ 是第${\displaystyle i}$維度的位置，此時 ${\displaystyle D^{1}}$為拖曳向量${\displaystyle D^{2}}$擴散張量

## 其他

${\displaystyle {\frac {\partial P}{\partial t}}=\nabla \cdot (P\nabla V)+D\nabla ^{2}P}$

${\displaystyle {\frac {\partial P}{\partial t}}=D\nabla ^{2}P}$

## 參考資料

1. ^ Leo P. Kadanoff. Statistical Physics: statics, dynamics and renormalization. World Scientific. 2000. ISBN 9810237642.
2. ^ A. D. Fokker, Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld, Ann. Phys. 348 (4. Folge 43), 810–820 (1914).
3. ^ M. Planck, Sitz.ber. Preuß. Akad. (1917).
4. ^ Edward Nelson ,"Derivation of the Schrödinger Equation from Newtonian Mechanics",Phys. Rev. 150, 1079–1085 (1966)