在 马尔可夫随机过程 中已经根据马尔可夫过程得到 Chapman-Kolmogorov 方程，这里再做概述。
Chapman-Kolmogorov 方程
根据马尔科夫过程的定义，系统的联合概率密度 (不同时刻处于不同状态的概率分布) 可以表示为：
p ( x 1 , t 1 ; y , t ; x 2 , t 2 ) = p ( x 2 , t 2 ∣ y , t ) p ( y , t ∣ x 1 , t 1 ) p ( x 1 , t 1 ) (1) p(x_{1},t_{1};y,t;x_{2},t_{2}) = p(x_{2},t_{2}\mid y,t) p(y,t\mid x_{1},t_{1}) p(x_{1},t_{1}) \tag{1} p ( x 1 , t 1 ; y , t ; x 2 , t 2 ) = p ( x 2 , t 2 ∣ y , t ) p ( y , t ∣ x 1 , t 1 ) p ( x 1 , t 1 ) ( 1 ) 要考虑 t 1 → t 2 t_{1}\to t_{2} t 1 → t 2 ( 1 ) (1) ( 1 ) y y y
p ( x 2 , t 2 ; x 1 , t 1 ) = p ( x 1 , t 1 ) ∫ p ( y , t ∣ x 1 , t 1 ) p ( x 2 , t 2 ∣ y , t ) d y (2) p(x_{2},t_{2};x_{1},t_{1}) = p(x_{1},t_{1}) \int p(y,t\mid x_{1},t_{1})p(x_{2},t_{2}\mid y,t) \, dy \tag{2} p ( x 2 , t 2 ; x 1 , t 1 ) = p ( x 1 , t 1 ) ∫ p ( y , t ∣ x 1 , t 1 ) p ( x 2 , t 2 ∣ y , t ) d y ( 2 ) 由条件概率的性质
p ( x 2 , t 2 ∣ x 1 , t 1 ) = p ( x 2 , t 2 ; x 1 , t 1 ) p ( x 1 , t 1 ) (3) p(x_{2},t_{2}\mid x_{1},t_{1}) = \frac{p(x_{2},t_{2};x_{1},t_{1})}{p(x_{1},t_{1})} \tag{3} p ( x 2 , t 2 ∣ x 1 , t 1 ) = p ( x 1 , t 1 ) p ( x 2 , t 2 ; x 1 , t 1 ) ( 3 ) 则有
p ( x 2 , t 2 ∣ x 1 , t 1 ) = ∫ p ( y , t ∣ x 1 , t 1 ) p ( x 2 , t 2 ∣ y , t ) d y (4) p(x_{2},t_{2}\mid x_{1},t_{1}) = \int p(y, t \mid x_{1},t_{1})p(x_{2},t_{2}\mid y,t) \, dy \tag{4} p ( x 2 , t 2 ∣ x 1 , t 1 ) = ∫ p ( y , t ∣ x 1 , t 1 ) p ( x 2 , t 2 ∣ y , t ) d y ( 4 ) 此式称为 Chapman-Kolmogorov 方程 。
∫ p ( x 2 , t 2 ∣ x 1 , t 1 ) d x 2 = 1 (5) \int p(x_{2},t_{2}\mid x_{1},t_{1}) \, dx_{2} =1 \tag{5} ∫ p ( x 2 , t 2 ∣ x 1 , t 1 ) d x 2 = 1 ( 5 )
如果 t 2 → t 1 t_{2}\to t_{1} t 2 → t 1
p ( x 2 , t 2 ∣ x 1 , t 1 ) = δ ( x 2 − x 1 ) (6) p(x_{2},t_{2}\mid x_{1},t_{1}) = \delta(x_{2}-x_{1}) \tag{6} p ( x 2 , t 2 ∣ x 1 , t 1 ) = δ ( x 2 − x 1 ) ( 6 ) From Chapman-Kolmogorov Equation To Master Equation
为了求解式 ( 4 ) (4) ( 4 )
{ ( x 1 , t 1 ) → ( x 0 , t 0 ) ( y , t ) → ( x ′ , t ) ( x 2 , t 2 ) → ( x , t + Δ t ) , (7) \begin{cases}
(x_{1},t_{1})\to (x_{0},t_{0}) \\ \\
(y,t) \to (x',t) \\ \\
(x_{2},t_{2}) \to (x,t+\Delta t),
\end{cases} \tag{7} ⎩ ⎨ ⎧ ( x 1 , t 1 ) → ( x 0 , t 0 ) ( y , t ) → ( x ′ , t ) ( x 2 , t 2 ) → ( x , t + Δ t ) , ( 7 ) 我们重写公式 ( 4 ) (4) ( 4 )
p ( x , t + Δ t ∣ x 0 , t 0 ) = ∫ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) d x ′ (8) p(x,t+\Delta t\mid x_{0},t_{0}) = \int p(x,t+\Delta t\mid x',t)p(x',t\mid x_{0},t_{0}) \, dx' \tag{8} p ( x , t + Δ t ∣ x 0 , t 0 ) = ∫ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) d x ′ ( 8 ) 我们现在引入 p ( x , t ∣ x 0 , t 0 ) p(x,t\mid x_{0},t_{0}) p ( x , t ∣ x 0 , t 0 ) t t t
∂ ∂ t p ( x ; t ∣ x 0 , t 0 ) = lim Δ t → 0 1 Δ t { [ p ( x , t + Δ t ∣ x 0 , t 0 ) − p ( x ; t ∣ x 0 , t 0 ) ] } (9) \frac{\partial}{\partial t}p(x;t\mid x_{0},t_{0}) = \lim_{ \Delta t \to 0 } \frac{1}{\Delta t}\left\{ \left[ p(x,t+\Delta t\mid x_{0},t_{0})-p(x;t\mid x_{0},t_{0}) \right] \right\} \tag{9} ∂ t ∂ p ( x ; t ∣ x 0 , t 0 ) = Δ t → 0 lim Δ t 1 { [ p ( x , t + Δ t ∣ x 0 , t 0 ) − p ( x ; t ∣ x 0 , t 0 ) ] } ( 9 ) 将公式 ( 8 ) (8) ( 8 ) ( 9 ) (9) ( 9 )
∂ ∂ t p ( x ; t ∣ x 0 , t 0 ) = = lim Δ t → 0 1 Δ t [ ∫ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) d x ′ − 1 ⋅ p ( x ; t ∣ x 0 , t 0 ) ] (10) \begin{aligned}
&\frac{\partial}{\partial t} p\left(x ; t \mid x_{0}, t_{0}\right)= \\
&=\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t}\left[\int p(x,t+\Delta t\mid x',t)p(x',t\mid x_{0},t_{0}) \, dx' - 1 \cdot p\left(x ; t \mid x_{0}, t_{0}\right)\right]
\end{aligned} \tag{10} ∂ t ∂ p ( x ; t ∣ x 0 , t 0 ) = = Δ t → 0 lim Δ t 1 [ ∫ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) d x ′ − 1 ⋅ p ( x ; t ∣ x 0 , t 0 ) ] ( 10 ) 注意到式 10 10 10 1 1 1 ( 6 ) (6) ( 6 )
∫ p ( x ′ , t + Δ t ∣ x ; t ) d x ′ = 1 (11) \int p(x',t+\Delta t\mid x;t) \, dx' = 1 \tag{11} ∫ p ( x ′ , t + Δ t ∣ x ; t ) d x ′ = 1 ( 11 ) 对于积分里面的条件积分我们利用公式 ( 6 ) (6) ( 6 )
∫ p ( x ′ , t + Δ t ∣ x ; t ) d x ′ = ∫ δ ( x ′ − x ) d x ′ \int p(x',t+\Delta t\mid x;t) \, dx' = \int \delta(x'-x) \, dx' ∫ p ( x ′ , t + Δ t ∣ x ; t ) d x ′ = ∫ δ ( x ′ − x ) d x ′ 显然该积分式等于 1 1 1 狄拉克函数
于是得到下式
∂ ∂ t p ( x ; t ∣ x 0 , t 0 ) = = lim Δ t → 0 1 Δ t [ ∫ Ω d x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − − ∫ Ω d x ′ p ( x ′ , t + Δ t ∣ x ; t ) p ( x ; t ∣ x 0 , t 0 ) ] . (12) \begin{aligned}
\frac{\partial}{\partial t} p\left(x ; t \mid x_{0}, t_{0}\right)&=\\
&=\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t}\left[\int_{\Omega} d x^{\prime} {\color{Red} p\left(x, t+\Delta t \mid x^{\prime}, t\right)} p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)-\right. \\
&\left.-\int_{\Omega} d x^{\prime} {\color{Red} p\left(x^{\prime}, t+\Delta t \mid x ; t\right)} p\left(x ; t \mid x_{0}, t_{0}\right)\right] .
\end{aligned} \tag{12} ∂ t ∂ p ( x ; t ∣ x 0 , t 0 ) = = Δ t → 0 lim Δ t 1 [ ∫ Ω d x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − − ∫ Ω d x ′ p ( x ′ , t + Δ t ∣ x ; t ) p ( x ; t ∣ x 0 , t 0 ) ] . ( 12 ) 我们引入项
W ( x , t ∣ x ′ , t ) = lim Δ t → 0 1 Δ t p ( x , t + Δ t ∣ x ′ , t ) W ( x ′ , t ∣ x , t ) = lim Δ t → 0 1 Δ t p ( x ′ , t + Δ t ∣ x , t ) , (13) \begin{split}
W(x,t\mid x',t) = \lim_{ \Delta t \to 0} \frac{1}{\Delta t}p(x,t+\Delta t\mid x',t)\\
W(x',t\mid x,t) = \lim_{ \Delta t \to 0 } \frac{1}{\Delta t}p(x',t+\Delta t\mid x,t),
\end{split} \tag{13} W ( x , t ∣ x ′ , t ) = Δ t → 0 lim Δ t 1 p ( x , t + Δ t ∣ x ′ , t ) W ( x ′ , t ∣ x , t ) = Δ t → 0 lim Δ t 1 p ( x ′ , t + Δ t ∣ x , t ) , ( 13 ) 这里 x ≠ x ′ x \ne x' x = x ′ W ( x , t ∣ x ′ , t ) W(x,t\mid x',t) W ( x , t ∣ x ′ , t ) W ( x ′ , t ∣ x , t ) W(x',t\mid x,t) W ( x ′ , t ∣ x , t ) [ time − 1 ] \left[ \text{time}^{-1} \right] [ time − 1 ]
至此，我们便得到主方程：
∂ ∂ t p ( x , t ∣ x 0 , t 0 ) = ∫ [ W ( x , t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − W ( x ′ , t ∣ x , t ) p ( x , t ∣ x 0 , t 0 ) ] d x ′ (14) \frac{\partial}{\partial t}p(x,t\mid x_{0},t_{0})=\int \left[ W(x,t\mid x',t)p(x',t\mid x_{0},t_{0})-W(x',t\mid x,t)p(x,t\mid x_{0},t_{0}) \right] \, dx' \tag{14} ∂ t ∂ p ( x , t ∣ x 0 , t 0 ) = ∫ [ W ( x , t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − W ( x ′ , t ∣ x , t ) p ( x , t ∣ x 0 , t 0 ) ] d x ′ ( 14 ) 写成离散的形式为：
∂ ∂ t p ( n , t ∣ n 0 , t 0 ) = ∑ n ′ [ W ( n , t ∣ n ′ , t ) p ( n ′ , t ∣ n 0 , t 0 ) − W ( n ′ , t ∣ n , t ) p ( n , t ∣ n 0 , t 0 ) ] . (15) \frac{\partial}{\partial t} p\left(n, t \mid n_{0}, t_{0}\right)=\sum_{n^{\prime}}\left[W\left(n, t \mid n^{\prime}, t\right) p\left(n^{\prime}, t \mid n_{0}, t_{0}\right)-W\left(n^{\prime}, t \mid n, t\right) p\left(n, t \mid n_{0}, t_{0}\right)\right] . \tag{15} ∂ t ∂ p ( n , t ∣ n 0 , t 0 ) = n ′ ∑ [ W ( n , t ∣ n ′ , t ) p ( n ′ , t ∣ n 0 , t 0 ) − W ( n ′ , t ∣ n , t ) p ( n , t ∣ n 0 , t 0 ) ] . ( 15 ) 主方程描述了系统处于状态 x x x
Kramers-Moyal 展开式的推导
我们对公式 ( 12 ) (12) ( 12 ) φ : Ω → R \varphi:\Omega\to \mathbb{R} φ : Ω → R x x x
∂ ∂ t ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = lim Δ t → 0 1 Δ t { ∫ Ω d x ∫ Ω d x ′ φ ( x ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − − ∫ Ω d x ∫ Ω d x ′ φ ( x ) p ( x ′ , t + Δ t ∣ x , t ) p ( x , t ∣ x 0 , t 0 ) } . (16) \begin{split}
&\frac{\partial}{\partial t} \int_{\Omega} d x \varphi(x) p\left(x, t \mid x_{0}, t_{0}\right)= \\
&=\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t}\left\{\int_{\Omega} d x \int_{\Omega} d x^{\prime} \varphi(x) p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)-\right. \\
&\left.-\int_{\Omega} d x \int_{\Omega} d x^{\prime} \varphi(x) p\left(x^{\prime}, t+\Delta t \mid x, t\right) p\left(x, t \mid x_{0}, t_{0}\right)\right\} .
\end{split} \tag{16} ∂ t ∂ ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = Δ t → 0 lim Δ t 1 { ∫ Ω d x ∫ Ω d x ′ φ ( x ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − − ∫ Ω d x ∫ Ω d x ′ φ ( x ) p ( x ′ , t + Δ t ∣ x , t ) p ( x , t ∣ x 0 , t 0 ) } . ( 16 ) 对于测试函数我们考虑在 x ′ x' x ′
φ ( x ) = φ ( x ′ ) + ∑ m = 1 ∞ ( x − x ′ ) m 1 m ! ∂ m φ ∂ x m ∣ x = x ′ . (17) \varphi(x)=\varphi\left(x^{\prime}\right)+\left.\sum_{m=1}^{\infty}\left(x-x^{\prime}\right)^{m} \frac{1}{m !} \frac{\partial^{m} \varphi}{\partial x^{m}}\right|_{x=x^{\prime}} . \tag{17} φ ( x ) = φ ( x ′ ) + m = 1 ∑ ∞ ( x − x ′ ) m m ! 1 ∂ x m ∂ m φ x = x ′ . ( 17 ) 我们把公式 ( 17 ) (17) ( 17 ) ( 16 ) (16) ( 16 )
∂ ∂ t ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = lim Δ t → 0 1 Δ t { ∫ Ω d x [ ∫ Ω d x ′ φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) + + ∫ Ω d x ′ ∑ m = 1 ∞ ( x − x ′ ) m 1 m ! ∂ m φ ∂ x m ∣ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − ] − ∫ Ω d x ∫ Ω d x ′ φ ( x ) p ( x ′ ; t + Δ t ∣ x ; t ) p ( x ; t ∣ x 0 , t 0 ) } . (18) \begin{split}
&\frac{\partial}{\partial t} \int_{\Omega} d x \varphi(x) p\left(x, t \mid x_{0}, t_{0}\right)= \\
&=\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t}\left\{\int _ { \Omega } d x \left[\int_{\Omega} d x^{\prime} {\color{Red}\varphi\left(x^{\prime}\right)} p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)+\right.\right. \\
&\left.+\left.\int_{\Omega} d x^{\prime} {\color{Red} \sum_{m=1}^{\infty}\left(x-x^{\prime}\right)^{m} \frac{1}{m !}} {\color{Red} \frac{\partial^{m} \varphi}{\partial x^{m}}}\right|_{\color{Red}{x=x^{\prime}}} p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)-\right] \\
&\left.-\int_{\Omega} d x \int_{\Omega} d x^{\prime} \varphi({\color{Blue}x}) p\left({\color{Blue}x^{\prime}} ; t+\Delta t \mid {\color{Blue}x} ; t\right) p\left({\color{Blue}x} ; t \mid x_{0}, t_{0}\right)\right\} .
\end{split} \tag{18} ∂ t ∂ ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = Δ t → 0 lim Δ t 1 { ∫ Ω d x [ ∫ Ω d x ′ φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) + + ∫ Ω d x ′ m = 1 ∑ ∞ ( x − x ′ ) m m ! 1 ∂ x m ∂ m φ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − ] − ∫ Ω d x ∫ Ω d x ′ φ ( x ) p ( x ′ ; t + Δ t ∣ x ; t ) p ( x ; t ∣ x 0 , t 0 ) } . ( 18 ) 这里的最后一个积分是在 Ω × Ω \Omega \times \Omega Ω × Ω
∂ ∂ t ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = lim Δ t → 0 1 Δ t { ∫ Ω d x [ ∫ Ω d x ′ φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) + + ∫ Ω d x ′ ∑ m = 1 ∞ ( x − x ′ ) m 1 m ! ∂ m φ ∂ x m ∣ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − ] − ∫ Ω d x ∫ Ω d x ′ φ ( x ′ ) p ( x ; t + Δ t ∣ x ′ ; t ) p ( x ′ ; t ∣ x 0 , t 0 ) } . (19) \begin{aligned}
& \frac{\partial}{\partial t} \int_{\Omega} d x \varphi(x) p\left(x, t \mid x_{0}, t_{0}\right)= \\
=\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t}\left\{\int_{\Omega} d x\right. & {\left[\int_{\Omega} d x^{\prime} \varphi\left(x^{\prime}\right) p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)+\right.} \\
& \left.+\left.\int_{\Omega} d x^{\prime} \sum_{m=1}^{\infty}\left(x-x^{\prime}\right)^{m} \frac{1}{m !} \frac{\partial^{m} \varphi}{\partial x^{m}}\right|_{x=x^{\prime}} p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)-\right] \\
- & \left.\int_{\Omega} d x \int_{\Omega} d x^{\prime} \varphi\left({\color{Blue}x^{\prime}}\right) p\left({\color{Blue}x} ; t+\Delta t \mid {\color{Blue}x^{\prime}} ; t\right) p\left({\color{Blue}x^{\prime}} ; t \mid x_{0}, t_{0}\right)\right\} .
\end{aligned} \tag{19} = Δ t → 0 lim Δ t 1 { ∫ Ω d x − ∂ t ∂ ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = [ ∫ Ω d x ′ φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) + + ∫ Ω d x ′ m = 1 ∑ ∞ ( x − x ′ ) m m ! 1 ∂ x m ∂ m φ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − ] ∫ Ω d x ∫ Ω d x ′ φ ( x ′ ) p ( x ; t + Δ t ∣ x ′ ; t ) p ( x ′ ; t ∣ x 0 , t 0 ) } . ( 19 ) 重新排列积分，我们有
∂ ∂ t ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = lim Δ t → 0 1 Δ t { ∫ Ω d x ′ [ ∫ Ω d x φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) ‾ + + ∫ Ω d x ∑ m = 1 ∞ ( x − x ′ ) m 1 m ! ∂ m φ ∂ x m ∣ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − − ∫ Ω d x φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) ‾ ] } . (20) \begin{split}
& \frac{\partial}{\partial t} \int_{\Omega} d x \varphi(x) p\left(x, t \mid x_{0}, t_{0}\right)= \\
& =\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t}\left\{\int _ { \Omega } d x ^ { \prime } \left[\int_{\Omega} d x \underline{\varphi\left(x^{\prime}\right) p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)}+\right.\right. \\
& +\left.\int_{\Omega} d x \sum_{m=1}^{\infty}\left(x-x^{\prime}\right)^{m} \frac{1}{m !} \frac{\partial^{m} \varphi}{\partial x^{m}}\right|_{x=x^{\prime}} p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)- \\
& \left.\left.-\int_{\Omega} d x \underline{\varphi\left(x^{\prime}\right) p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)}\right]\right\} . \\
\end{split} \tag{20} ∂ t ∂ ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = Δ t → 0 lim Δ t 1 { ∫ Ω d x ′ [ ∫ Ω d x φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) + + ∫ Ω d x m = 1 ∑ ∞ ( x − x ′ ) m m ! 1 ∂ x m ∂ m φ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) − − ∫ Ω d x φ ( x ′ ) p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) ] } . ( 20 ) 我们有：
∂ ∂ t ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = lim Δ t → 0 1 Δ t ‾ { ∫ Ω d x ′ ∫ Ω d x ∑ m = 1 ∞ ( x − x ′ ) m 1 m ! ∂ m φ ∂ x m ∣ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) } . (21) \begin{split}
& \frac{\partial}{\partial t} \int_{\Omega} d x \varphi(x) p\left(x, t \mid x_{0}, t_{0}\right)= \\
& =\underline{\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t}}\left\{\left.\int_{\Omega} d x^{\prime} \int_{\Omega} d x \sum_{m=1}^{\infty}\left(x-x^{\prime}\right)^{m} \frac{1}{m !} \frac{\partial^{m} \varphi}{\partial x^{m}}\right|_{x=x^{\prime}} p\left(x, t+\Delta t \mid x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)\right\} .
\end{split} \tag{21} ∂ t ∂ ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = Δ t → 0 lim Δ t 1 { ∫ Ω d x ′ ∫ Ω d x m = 1 ∑ ∞ ( x − x ′ ) m m ! 1 ∂ x m ∂ m φ x = x ′ p ( x , t + Δ t ∣ x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) } . ( 21 ) 我们定义系数
D m ( x ′ , t ) = lim Δ t → 0 1 Δ t ∫ Ω d x ( x − x ′ ) m p ( x , t + Δ t ∣ x ′ , t ) , (22) D_{m}\left(x^{\prime}, t\right)=\lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t} \int_{\Omega} d x\left(x-x^{\prime}\right)^{m} p\left(x, t+\Delta t \mid x^{\prime}, t\right), \tag{22} D m ( x ′ , t ) = Δ t → 0 lim Δ t 1 ∫ Ω d x ( x − x ′ ) m p ( x , t + Δ t ∣ x ′ , t ) , ( 22 ) 那式子 ( 21 ) (21) ( 21 )
∂ ∂ t ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = ∫ Ω d x ′ [ ∑ m = 1 ∞ D m ( x ′ , t ) 1 m ! ∂ m φ ∂ x m ∣ x = x ′ p ( x ′ , t ∣ x 0 , t 0 ) ] . (23) \begin{split}
\frac{\partial}{\partial t} \int_{\Omega} d x \varphi(x) p\left(x, t \mid x_{0}, t_{0}\right) & = \\
& =\int_{\Omega} d x^{\prime}\left[\left.\sum_{m=1}^{\infty} D_{m}\left(x^{\prime}, t\right) \frac{1}{m !} \frac{\partial^{m} \varphi}{\partial x^{m}}\right|_{x=x^{\prime}} p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)\right] .
\end{split} \tag{23} ∂ t ∂ ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = ∫ Ω d x ′ [ m = 1 ∑ ∞ D m ( x ′ , t ) m ! 1 ∂ x m ∂ m φ x = x ′ p ( x ′ , t ∣ x 0 , t 0 ) ] . ( 23 ) 于是我们得到
∂ ∂ t ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = ∑ m = 1 ∞ ( − 1 ) m m ! ∫ Ω d x ′ [ φ ( x ′ ) ∂ m ∂ x m [ D m ( x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) ] ] . \begin{split}
&\frac{\partial}{\partial t} \int_{\Omega} d x \varphi(x) p\left(x, t \mid x_{0}, t_{0}\right)= \\
&=\sum_{m=1}^{\infty} \frac{(-1)^{m}}{m !} \int_{\Omega} d x^{\prime}\left[\varphi\left(x^{\prime}\right) \frac{\partial^{m}}{\partial x^{m}}\left[D_{m}\left(x^{\prime}, t\right) p\left(x^{\prime}, t \mid x_{0}, t_{0}\right)\right]\right] .
\end{split} ∂ t ∂ ∫ Ω d x φ ( x ) p ( x , t ∣ x 0 , t 0 ) = = m = 1 ∑ ∞ m ! ( − 1 ) m ∫ Ω d x ′ [ φ ( x ′ ) ∂ x m ∂ m [ D m ( x ′ , t ) p ( x ′ , t ∣ x 0 , t 0 ) ] ] . 因为上式对于任何测试函数 φ \varphi φ
∂ ∂ t p ( x , t ∣ x 0 , t 0 ) = ∑ m = 1 ∞ ( − 1 ) m m ! ∂ m ∂ x m [ D m ( x , t ) p ( x , t ∣ x 0 , t 0 ) ] \frac{\partial}{\partial t} p\left(x, t \mid x_{0}, t_{0}\right)=\sum_{m=1}^{\infty} \frac{(-1)^{m}}{m !} \frac{\partial^{m}}{\partial x^{m}}\left[D_{m}(x, t) p\left(x, t \mid x_{0}, t_{0}\right)\right] ∂ t ∂ p ( x , t ∣ x 0 , t 0 ) = m = 1 ∑ ∞ m ! ( − 1 ) m ∂ x m ∂ m [ D m ( x , t ) p ( x , t ∣ x 0 , t 0 ) ] 这就是著名 Kramers-Moyal Expansion(KME)
The Fokker-Planck Equation
通过截断 KME 到第二项，我们得到 Fokker-Planck Equation
∂ p ( x , t ) ∂ t = − ∂ ∂ x [ D 1 ( x , t ) p ( x , t ) ] + 1 2 ∂ 2 ∂ x 2 [ D 2 ( x , t ) p ( x , t ) ] . \frac{\partial p(x, t)}{\partial t}=-\frac{\partial}{\partial x}\left[D_{1}(x, t) p(x, t)\right]+\frac{1}{2} \frac{\partial^{2}}{\partial x^{2}}\left[D_{2}(x, t) p(x, t)\right] . ∂ t ∂ p ( x , t ) = − ∂ x ∂ [ D 1 ( x , t ) p ( x , t ) ] + 2 1 ∂ x 2 ∂ 2 [ D 2 ( x , t ) p ( x , t ) ] . 其中 D 1 ( x , t ) D_{1}(x,t) D 1 ( x , t ) D 2 ( x , t ) D_{2}(x,t) D 2 ( x , t )
相关资料
