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Chapman-Kolmogorov 方程到福克普郎克方程 (FPK 方程)

· 8 min read
PuQing
AI, CVer, Pythoner, Half-stack Developer

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(x1,t1;y,t;x2,t2)=p(x2,t2y,t)p(y,tx1,t1)p(x1,t1)(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}

要考虑 t1t2t_{1}\to t_{2} 所有中间过程。则对上式 (1)(1)yy 求积分,有:

p(x2,t2;x1,t1)=p(x1,t1)p(y,tx1,t1)p(x2,t2y,t)dy(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}

由条件概率的性质

p(x2,t2x1,t1)=p(x2,t2;x1,t1)p(x1,t1)(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}

则有

p(x2,t2x1,t1)=p(y,tx1,t1)p(x2,t2y,t)dy(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}

此式称为 Chapman-Kolmogorov 方程

CKE 性质
  • CKE 满足归一条件
p(x2,t2x1,t1)dx2=1(5)\int p(x_{2},t_{2}\mid x_{1},t_{1}) \, dx_{2} =1 \tag{5}
  • 如果 t2t1t_{2}\to t_{1},则
p(x2,t2x1,t1)=δ(x2x1)(6)p(x_{2},t_{2}\mid x_{1},t_{1}) = \delta(x_{2}-x_{1}) \tag{6}

From Chapman-Kolmogorov Equation To Master Equation

为了求解式 (4)(4) 中的条件概率随时间的变化,我们需要引入微分方程。为此,我们对符号进行如下修改:

{(x1,t1)(x0,t0)(y,t)(x,t)(x2,t2)(x,t+Δt),(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}

我们重写公式 (4)(4)

p(x,t+Δtx0,t0)=p(x,t+Δtx,t)p(x,tx0,t0)dx(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}

我们现在引入 p(x,tx0,t0)p(x,t\mid x_{0},t_{0}) 对于 tt 的微分

tp(x;tx0,t0)=limΔt01Δt{[p(x,t+Δtx0,t0)p(x;tx0,t0)]}(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}

将公式 (8)(8) 代入到公式 (9)(9)

tp(x;tx0,t0)==limΔt01Δt[p(x,t+Δtx,t)p(x,tx0,t0)dx1p(x;tx0,t0)](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}

注意到式 1010 积分右手项是乘以 11,我们可以利用式 (6)(6) 来改造这样一个积分

p(x,t+Δtx;t)dx=1(11)\int p(x',t+\Delta t\mid x;t) \, dx' = 1 \tag{11}
证明

对于积分里面的条件积分我们利用公式 (6)(6) 得到

p(x,t+Δtx;t)dx=δ(xx)dx\int p(x',t+\Delta t\mid x;t) \, dx' = \int \delta(x'-x) \, dx'

显然该积分式等于 11,见 狄拉克函数

于是得到下式

tp(x;tx0,t0)==limΔt01Δt[Ωdxp(x,t+Δtx,t)p(x,tx0,t0)Ωdxp(x,t+Δtx;t)p(x;tx0,t0)].(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}

我们引入项

W(x,tx,t)=limΔt01Δtp(x,t+Δtx,t)W(x,tx,t)=limΔt01Δtp(x,t+Δtx,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}

这里 xxx \ne x'。这里 W(x,tx,t)W(x,t\mid x',t)W(x,tx,t)W(x',t\mid x,t) 的单位是 [time1]\left[ \text{time}^{-1} \right],并称为转移率。

至此,我们便得到主方程:

tp(x,tx0,t0)=[W(x,tx,t)p(x,tx0,t0)W(x,tx,t)p(x,tx0,t0)]dx(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}

写成离散的形式为:

tp(n,tn0,t0)=n[W(n,tn,t)p(n,tn0,t0)W(n,tn,t)p(n,tn0,t0)].(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}

主方程描述了系统处于状态 xx 的概率随时间的变化率,等于单位时间流入该点的机率流减去单位时间内由该点流出的机率流。即:系统在一段时间内发生的使系统离开原来状态的行为的概率减该时间段内没有发生使系统离开原来状态的概率,或表示单位时间内粒子数的变化等于别处粒子向该处补充的数目减去该处粒子向周围移走的数目。

Kramers-Moyal 展开式的推导

我们对公式 (12)(12) 两边同时乘以测试函数 φ:ΩR\varphi:\Omega\to \mathbb{R} 并对 xx 进行积分

tΩdxφ(x)p(x,tx0,t0)==limΔt01Δt{ΩdxΩdxφ(x)p(x,t+Δtx,t)p(x,tx0,t0)ΩdxΩdxφ(x)p(x,t+Δtx,t)p(x,tx0,t0)}.(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}

对于测试函数我们考虑在 xx' 进行泰勒展开

φ(x)=φ(x)+m=1(xx)m1m!mφxmx=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}

我们把公式 (17)(17) 回代到公式 (16)(16) 第一个积分项

tΩdxφ(x)p(x,tx0,t0)==limΔt01Δt{Ωdx[Ωdxφ(x)p(x,t+Δtx,t)p(x,tx0,t0)++Ωdxm=1(xx)m1m!mφxmx=xp(x,t+Δtx,t)p(x,tx0,t0)]ΩdxΩdxφ(x)p(x;t+Δtx;t)p(x;tx0,t0)}.(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}

这里的最后一个积分是在 Ω×Ω\Omega \times \Omega 上的二重积分,并且积分顺序无关,所以我们交换积分变量

tΩdxφ(x)p(x,tx0,t0)==limΔt01Δt{Ωdx[Ωdxφ(x)p(x,t+Δtx,t)p(x,tx0,t0)++Ωdxm=1(xx)m1m!mφxmx=xp(x,t+Δtx,t)p(x,tx0,t0)]ΩdxΩdxφ(x)p(x;t+Δtx;t)p(x;tx0,t0)}.(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}

重新排列积分,我们有

tΩdxφ(x)p(x,tx0,t0)==limΔt01Δt{Ωdx[Ωdxφ(x)p(x,t+Δtx,t)p(x,tx0,t0)++Ωdxm=1(xx)m1m!mφxmx=xp(x,t+Δtx,t)p(x,tx0,t0)Ωdxφ(x)p(x,t+Δtx,t)p(x,tx0,t0)]}.(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}

我们有:

tΩdxφ(x)p(x,tx0,t0)==limΔt01Δt{ΩdxΩdxm=1(xx)m1m!mφxmx=xp(x,t+Δtx,t)p(x,tx0,t0)}.(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}

我们定义系数

Dm(x,t)=limΔt01ΔtΩdx(xx)mp(x,t+Δtx,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}

那式子 (21)(21) 就有

tΩdxφ(x)p(x,tx0,t0)==Ωdx[m=1Dm(x,t)1m!mφxmx=xp(x,tx0,t0)].(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}

于是我们得到

tΩdxφ(x)p(x,tx0,t0)==m=1(1)mm!Ωdx[φ(x)mxm[Dm(x,t)p(x,tx0,t0)]].\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}

因为上式对于任何测试函数 φ\varphi 都成立,所以得到

tp(x,tx0,t0)=m=1(1)mm!mxm[Dm(x,t)p(x,tx0,t0)]\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]

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[D1(x,t)p(x,t)]+122x2[D2(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] .

其中 D1(x,t)D_{1}(x,t) 是漂移,D2(x,t)D_{2}(x,t) 是扩散

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