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] 荷兰 物理 学家阿德里安·福克 [ 2] 马克斯·普朗克 [ 3]
存在拖拽和扩散项时，福克-普朗克方程的一个一维解。初始状态为远离零速度的δ函数，随机冲击使其分布逐渐变宽 一维 x 方向上,福克-普朗克方程有两个参数，一是拖拽参数 D 1 (x ,t )，另一是扩散 D 2 (x ,t )
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{\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].}
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{\displaystyle N}
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{\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],}
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{\displaystyle x_{i}}
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{\displaystyle D^{1}}
矢量 ，
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{\displaystyle D^{2}}
扩散 张量 。
![{\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].}](http://wikimedia.org/api/rest_v1/media/math/render/svg/a851f9f8af87ef864a937f4fd0eae85f750c549b)
![{\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],}](http://wikimedia.org/api/rest_v1/media/math/render/svg/619522eefdacb8073c9d778fe21d3bd0c0285fe0)
