福克-普朗克方程

福克-普朗克方程（Fokker–Planck equation）描述粒子在势能场中受到随机力后，随时间演化的位置或是速度的分布函数 ^[1] 。此方程以荷兰物理学家阿德里安·福克^[2]与马克斯·普朗克^[3]的姓氏来命名。

一维 x方向上,福克-普朗克方程有两个参数，一是拖拽参数 D₁(x,t)，另一是扩散 D₂(x,t)

{\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].

在 $N$ 维空间中的福克-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

维度的位置，此时

D^{1}

为拖拽矢量，

D^{2}

其他

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

若V=0，则福克-普朗克方程成为布朗运动

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

参考资料

[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)

延伸阅读

Hannes Risken, "The Fokker–Planck equation : Methods of Solutions and Applications", 2nd edition, Springer Series in Synergetics, Springer, ISBN 3-540-61530-X.
David Tong. Kinetic Theory. Ch. 3. https://www.damtp.cam.ac.uk/user/tong/kinetic.html （页面存档备份，存于互联网档案馆）
Scott. Applied Stochastic Processes.