# 福克-普朗克方程

（重定向自佛客-普朗克方程式

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

$N$ 空間中的福克-普朗克方程是

$\frac{\partial f}{\partial t} = -\sum_{i=1}^N \frac{\partial}{\partial x_i} \left[ D_i^1(x_1, \ldots, x_N) f \right] + \sum_{i=1}^{N} \sum_{j=1}^{N} \frac{\partial^2}{\partial x_i \, \partial x_j} \left[ D_{ij}^2(x_1, \ldots, x_N) f \right],$ $x_i$ 是第$i$維度的位置，此時 $D^1$為拖曳向量$D^2$擴散張量

## 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)

## 延伸閱讀

• Hannes Risken, "The Fokker–Planck equation : Methods of Solutions and Applications", 2nd edition, Springer Series in Synergetics, Springer, ISBN 3-540-61530-X.