CTRW的化学主方程推导(下)

书接上回:
CTRW的化学主方程推导(上)

定义

m s m_{s} ms:代表不同物种的个数
m r m_{r} mr:代表不同反应的个数,物种与反应共同组成一个反应系统。
S j S_{j} Sj:代表不同的物种, j j j的取值从1到 m s m_{s} ms
n j n_{j} nj:物种 S j S_{j} Sj对应的粒子数
n = ( n 1 , ⋯   , n m s ) T \mathbf{n}=\left(n_{1}, \cdots, n_{m_{s}}\right)^{\mathrm{T}} n=(n1,,nms)T:粒子数目的状态向量
r i j ∈ N ( p i j ∈ N ) r_{i j} \in \mathbb{N}\left(p_{i j} \in \mathbb{N}\right) rijN(pijN):经过反应 i i i n j n_{j} nj的变化量。
s j ˉ = p i j − r i j s_{\bar{j}}=p_{i j}-r_{i j} sjˉ=pijrij:化学计量系数,代表物种 j j j经过反应 i i i后的粒子数目变化
根据以上定义,我们可以把反应 i i i对于状态空间的影响表示为 ∑ j r i j S j → ∑ j p i j S j \sum_{j} r_{i j} S_{j} \rightarrow \sum_{j} p_{i j} S_{j} jrijSjjpijSj

推广的化学主方程

上一回我们得到了这条式子:
P ( n , t ) = ∫ 0 t ∑ k = 0 ∞ R k ( n , t ′ ) ∑ i = 1 m r ∫ t − t ′ ∞ ϕ i ( t ′ ′ ; n ) d t ′ ′ d t ′ P(\mathbf{n}, t)=\int_{0}^{t} \sum_{k=0}^{\infty} R_{k}\left(\mathbf{n}, t^{\prime}\right) \sum_{i=1}^{m_{r}} \int_{t-t^{\prime}}^{\infty} \phi_{i}\left(t^{\prime \prime} ; \mathbf{n}\right) d t^{\prime \prime} d t^{\prime} P(n,t)=0tk=0Rk(n,t)i=1mrttϕi(t;n)dtdt
其中, R k ( n , t ) = ⟨ δ n , N k δ ( T k − t ) ⟩ R_{k}(\mathbf{n}, t)=\left\langle\delta_{\mathbf{n}, \mathbf{N}_{k}} \delta\left(T_{k}-t\right)\right\rangle Rk(n,t)=δn,Nkδ(Tkt)是经历 k k k步反应后在 t t t时刻状态为 n \mathbf{n} n的联合密度, R ( n , t ) = ∑ k = 0 ∞ R k ( n , t ) R(\mathbf{n}, t)=\sum_{k=0}^{\infty} R_{k}(\mathbf{n}, t) R(n,t)=k=0Rk(n,t)是经历任意步反应后在 t t t时刻状态为 n \mathbf{n} n的概率密度。此外,我们还得到了一些递增关系。

那么接下来我们就继续从这里入手,递增关系暗含了未来的状态只取决于当前的状态,与过去历史的状态无关,所以这是一个关于步数 k k k的马尔科夫过程,对应的联合概率密度就是 R k ( n , t ) R_{k}(\mathbf{n}, t) Rk(n,t)满足 Chapamn-Kolmogorov 方程:
R k + 1 ( n , t ) = ∫ 0 t ∑ i = 1 m r R k ( n − s i , t ′ ) ϕ i ( t − t ′ ; n − s i ) d t ′ R_{k+1}(\mathbf{n}, t)=\int_{0}^{t} \sum_{i=1}^{m_{r}} R_{k}\left(\mathbf{n}-\mathbf{s}_{i}, t^{\prime}\right) \phi_{i}\left(t-t^{\prime} ; \mathbf{n}-\mathbf{s}_{i}\right) d t^{\prime} Rk+1(n,t)=0ti=1mrRk(nsi,t)ϕi(tt;nsi)dt
继续记 R 0 ( n , t ) = ⟨ δ n , N 0 δ ( T 0 − t ) ⟩ = P ( n , 0 ) δ ( t ) R_{0}(\mathbf{n}, t)=\left\langle\delta_{\mathbf{n}, \mathbf{N}_{0}} \delta\left(T_{0}-t\right)\right\rangle=P(\mathbf{n}, 0) \delta(t) R0(n,t)=δn,N0δ(T0t)=P(n,0)δ(t),注意到 P ( n , t ) P(\mathbf{n}, t) P(n,t)的表达式以及Chapamn-Kolmogorov方程都是卷积形式,所以我们可以使用Laplace 变换得到:
R ~ ( n , λ ) = P ( n , 0 ) + ∑ i = 1 m r R ~ ( n − s i , λ ) ϕ ~ i ( λ ; n − s i ) P ~ ( n , λ ) = R ~ ( n , λ ) 1 − ∑ i = 1 m r ϕ ~ i ( λ ; n ) λ \tilde{R}(\mathbf{n}, \lambda)=P(\mathbf{n}, 0)+\sum_{i=1}^{m_{r}} \tilde{R}\left(\mathbf{n}-\mathbf{s}_{i}, \lambda\right) \tilde{\phi}_{i}\left(\lambda ; \mathbf{n}-\mathbf{s}_{i}\right) \\ \tilde{P}(\mathbf{n}, \lambda)=\tilde{R}(\mathbf{n}, \lambda) \frac{1-\sum_{i=1}^{m_{r}} \tilde{\phi}_{i}(\lambda ; \mathbf{n})}{\lambda} R~(n,λ)=P(n,0)+i=1mrR~(nsi,λ)ϕ~i(λ;nsi)P~(n,λ)=R~(n,λ)λ1i=1mrϕ~i(λ;n)
联立R与P可以得到:
λ P ~ ( n , λ ) = [ P ( n , 0 ) + ∑ i = 1 m r R ~ ( n − s i , λ ) ϕ ~ i ( λ ; n − s i ) ] ( 1 − ∑ i = 1 m r ϕ ~ i ( λ ; n ) ) \lambda \tilde{P}(\mathbf{n}, \lambda)=\left[P(\mathbf{n}, 0)+\sum_{i=1}^{m_{r}} \tilde{R}\left(\mathbf{n}-\mathbf{s}_{i}, \lambda\right) \widetilde{\phi}_{i}\left(\lambda ; \mathbf{n}-\mathbf{s}_{i}\right)\right]\left(1-\sum_{i=1}^{m_{r}} \tilde{\phi}_{i}(\lambda ; \mathbf{n})\right) λP~(n,λ)=[P(n,0)+i=1mrR~(nsi,λ)ϕ i(λ;nsi)](1i=1mrϕ~i(λ;n))
又因为:
P ~ ( n , λ ) M ~ i ( λ ; n ) = R ~ ( n , λ ) ϕ ~ i ( λ ; n ) ∑ i = 1 m r ϕ ~ i ( λ ; n ) [ P ( n , 0 ) + ∑ i = 1 m r R ~ ( n − s i , λ ) ϕ ~ i ( λ ; n − s i ) ] = ∑ i = 1 m r ϕ ~ i ( λ ; n ) R ~ ( n , λ ) = ∑ i = 1 m r P ~ ( n , λ ) M ~ i ( λ ; n ) \tilde{P}(\mathbf{n}, \lambda) \widetilde{M}_{i}(\lambda ; \mathbf{n})=\tilde{R}(\mathbf{n}, \lambda) \widetilde{\phi}_{i}(\lambda ; \mathbf{n}) \sum_{i=1}^{m_{r}} \widetilde{\phi}_{i}(\lambda ; \mathbf{n})\left[P(\mathbf{n}, 0)+\sum_{i=1}^{m_{r}} \tilde{R}\left(\mathbf{n}-\mathbf{s}_{i}, \lambda\right) \widetilde{\phi}_{i}\left(\lambda ; \mathbf{n}-\mathbf{s}_{i}\right)\right] \\ =\sum_{i=1}^{m_{r}} \widetilde{\phi}_{i}(\lambda ; \mathbf{n}) \tilde{R}(\mathbf{n}, \lambda) \\ =\sum_{i=1}^{m_{r}} \tilde{P}(\mathbf{n}, \lambda) \widetilde{M}_{i}(\lambda ; \mathbf{n}) P~(n,λ)M i(λ;n)=R~(n,λ)ϕ i(λ;n)i=1mrϕ i(λ;n)[P(n,0)+i=1mrR~(nsi,λ)ϕ i(λ;nsi)]=i=1mrϕ i(λ;n)R~(n,λ)=i=1mrP~(n,λ)M i(λ;n)
其中
M ~ i ( λ ; n ) = λ ϕ ~ i ( λ ; n ) 1 − ∑ l = 1 m r ϕ ~ l ( λ ; n ) \widetilde{M}_{i}(\lambda ; \mathbf{n})=\frac{\lambda \widetilde{\phi}_{i}(\lambda ; \mathbf{n})}{1-\sum_{l=1}^{m_{r}} \widetilde{\phi}_{l}(\lambda ; \mathbf{n})} M i(λ;n)=1l=1mrϕ l(λ;n)λϕ i(λ;n)
所以
λ P ~ ( n , λ ) = P ( n , 0 ) + ∑ i = 1 m r ( P ~ ( n − s i , λ ) M ~ i ( λ ; n − s i ) − P ~ ( n , λ ) M ~ i ( λ ; n ) ) = P ( n , 0 ) + ∑ i = 1 m ( ∏ j = 1 m i E j − s j − 1 ) P ~ ( n , λ ) M ~ i ( λ ; n ) \begin{aligned} \lambda \tilde{P}(\mathbf{n}, \lambda) &=P(\mathbf{n}, 0)+\sum_{i=1}^{m_{r}}\left(\tilde{P}\left(\mathbf{n}-\mathbf{s}_{i}, \lambda\right) \widetilde{M}_{i}\left(\lambda ; \mathbf{n}-\mathbf{s}_{i}\right)-\tilde{P}(\mathbf{n}, \lambda) \widetilde{M}_{i}(\lambda ; \mathbf{n})\right) \\ &=P(\mathbf{n}, 0)+\sum_{i=1}^{m}\left(\prod_{j=1}^{m_{i}} \mathrm{E}_{j}^{-s_{j}}-1\right) \tilde{P}(\mathbf{n}, \lambda) \widetilde{M}_{i}(\lambda ; \mathbf{n}) \end{aligned} λP~(n,λ)=P(n,0)+i=1mr(P~(nsi,λ)M i(λ;nsi)P~(n,λ)M i(λ;n))=P(n,0)+i=1m(j=1miEjsj1)P~(n,λ)M i(λ;n)
其中 E j z ( z ∈ Z ) \mathrm{E}_{j}^{z}(z \in \mathbb{Z}) Ejz(zZ)的定义为: E j z f ( n ) = f ( n 1 , … , n j + z , … , n m z ) \mathrm{E}_{j}^{z} f(\mathbf{n})=f\left(n_{1}, \ldots, n_{j}+z, \ldots, n_{m_{z}}\right) Ejzf(n)=f(n1,,nj+z,,nmz)。最后再做 Laplace逆变换,这样我们就能得到推广的化学主方程:
∂ P ( n , t ) ∂ t = ∑ i = 1 m r ∫ 0 t ( ∏ j = 1 m i E j − s i j − 1 ) P ( n , t ′ ) M i ( t − t ′ ; n ) d t ′ \frac{\partial P(\mathbf{n}, t)}{\partial t}=\sum_{i=1}^{m_{r}} \int_{0}^{t}\left(\prod_{j=1}^{m_{i}} \mathrm{E}_{j}^{-s_{i j}}-1\right) P\left(\mathbf{n}, t^{\prime}\right) M_{i}\left(t-t^{\prime} ; \mathbf{n}\right) d t^{\prime} tP(n,t)=i=1mr0t(j=1miEjsij1)P(n,t)Mi(tt;n)dt
其中记忆函数 M i ( t ; n ) M_{i}(t ; \mathbf{n}) Mi(t;n)我们是通过它的 Laplace 变换 M ~ i ( λ ; n ) \widetilde{M}_{i}(\lambda ; \mathbf{n}) M i(λ;n)来定义的。
至此我们已经完成了CTRW下化学主方程的推导,下面是一些其它延伸。

推广的化学速率方程

当粒子数目足够大的时候,我们可以考虑更便利的系统宏观行为,此时主方程可以 近似为一阶矩的形式。依然先给出一些定义。

定义

C = N / n 0 \mathbf{C}=\mathbf{N} / n_{0} C=N/n0:浓度
n 0 = ∑ j n j , 0 n_{0}=\sum_{j} n_{j, 0} n0=jnj,0:浓度中 n 0 n_{0} n0需满足的条件
P C ( c , t ) = n 0 P ( n 0 c , t ) P^{C}(\mathbf{c}, t)=n_{0} P\left(n_{0} \mathbf{c}, t\right) PC(c,t)=n0P(n0c,t) M i C ( t ; c ) = M i ( t ; n 0 c ) / n 0 M_{i}^{C}(t ; \mathbf{c})=M_{i}\left(t ; n_{0} \mathbf{c}\right) / n_{0} MiC(t;c)=Mi(t;n0c)/n0:将化学主方程中的差分近似为微分的前置条件。

推导

由上面的定义,考虑算子 E j z ( z ∈ Z ) \mathrm{E}_{j}^{z}(z \in \mathbb{Z}) Ejz(zZ),有:
( ∏ j = 1 m z E j − s i − 1 ) P ( u , t ′ ) M i ( t − t ′ ; n ) = P ( n − s i , t ′ ) M i ( t − t ′ ; n − s i ) − P ( n , t ′ ) M i ( t − t ′ ; n ) = − s i / n 0 s i / n 0 [ P c ( c , t ′ ) M i c ( t − t ′ ; c ) − P c ( c − s i n 0 , t ′ ) M i c ( t − t ′ ; c − s i n 0 ) ] ≈ − s i n 0 ⋅ ∇ P C ( c , t ′ ) M i C ( t − t ′ ; c ) \left(\prod_{j=1}^{m_{z}} \mathrm{E}_{j}^{-s_{i}}-1\right) P\left(\mathbf{u}, t^{\prime}\right) M_{i}\left(t-t^{\prime} ; \mathbf{n}\right) \\ =P\left(\mathbf{n}-\mathbf{s}_{i}, t^{\prime}\right) M_{i}\left(t-t^{\prime} ; \mathbf{n}-\mathbf{s}_{i}\right)-P\left(\mathbf{n}, t^{\prime}\right) M_{i}\left(t-t^{\prime} ; \mathbf{n}\right) \\ =-\frac{\mathbf{s}_{i} / n_{0}}{\mathbf{s}_{i} / n_{0}}\left[P^{c}\left(\mathbf{c}, t^{\prime}\right) M_{i}^{c}\left(t-t^{\prime} ; \mathbf{c}\right)-P^{c}\left(\mathbf{c}-\frac{\mathbf{s}_{i}}{n_{0}}, t^{\prime}\right) M_{i}^{c}\left(t-t^{\prime} ; \mathbf{c}-\frac{\mathbf{s}_{i}}{n_{0}}\right)\right] \\ \approx-\frac{\mathbf{s}_{i}}{n_{0}} \cdot \nabla P^{C}\left(\mathbf{c}, t^{\prime}\right) M_{i}^{C}\left(t-t^{\prime} ; \mathbf{c}\right) (j=1mzEjsi1)P(u,t)Mi(tt;n)=P(nsi,t)Mi(tt;nsi)P(n,t)Mi(tt;n)=si/n0si/n0[Pc(c,t)Mic(tt;c)Pc(cn0si,t)Mic(tt;cn0si)]n0siPC(c,t)MiC(tt;c)
类似地,同样地对于大粒子数目,我们有 ∑ n ≈ n 0 ∫ d c \sum_{\mathrm{n}} \approx n_{0} \int d \mathbf{c} nn0dc,因此在推广的化学主方程等式两边先乘上 n n n,对 n n n求和,并利用上述两个近似, 对 c c c进行分部积分,可以得到:
∑ n n ∂ P ( n , t ) ∂ t = − ∑ n n ∑ i = 1 m r ∫ 0 t s i n 0 ⋅ ∇ P C ( c , t ′ ) M i C ( t − t ′ ; c ) d t ′ ≈ − n 0 ∑ i = 1 m 2 s i ∫ 0 t ∫ 0 ∞ c ∇ P C ( c , t ′ ) M i C ( t − t ′ ; c ) d c d t ′ = n 0 ∑ i = 1 m r s i ∫ 0 ∞ ⟨ M i c [ t − t ′ ; C ( t ′ ) ] ⟩ d t ′ \begin{aligned} \sum_{\mathbf{n}} \mathbf{n} \frac{\partial P(\mathbf{n}, t)}{\partial t}=&-\sum_{\mathbf{n}} \mathbf{n} \sum_{i=1}^{m_{r}} \int_{0}^{t} \frac{\mathbf{s}_{i}}{n_{0}} \cdot \nabla P^{C}\left(\mathbf{c}, t^{\prime}\right) M_{i}^{C}\left(t-t^{\prime} ; \mathbf{c}\right) d t^{\prime} \\ & \approx-n_{0} \sum_{i=1}^{m_{2}} \mathbf{s}_{i} \int_{0}^{t} \int_{0}^{\infty} \mathbf{c} \nabla P^{C}\left(\mathbf{c}, t^{\prime}\right) M_{i}^{C}\left(t-t^{\prime} ; \mathbf{c}\right) d \mathbf{c} d t^{\prime} \\ =& n_{0} \sum_{i=1}^{m_{r}} \mathbf{s}_{i} \int_{0}^{\infty}\left\langle M_{i}^{c}\left[t-t^{\prime} ; \mathbf{C}\left(t^{\prime}\right)\right]\right\rangle d t^{\prime} \end{aligned} nntP(n,t)==nni=1mr0tn0siPC(c,t)MiC(tt;c)dtn0i=1m2si0t0cPC(c,t)MiC(tt;c)dcdtn0i=1mrsi0Mic[tt;C(t)]dt
又因为:
∑ n n ∂ P ( n , t ) ∂ t ≈ n 0 ∫ 0 ∞ n 0 C P ( n 0 c , t ) d c = n 0 ∫ 0 ∞ c P c ( c , t ) d c = n 0 ∂ ⟨ C ⟩ ∂ t \sum_{\mathbf{n}} \mathbf{n} \frac{\partial P(\mathbf{n}, t)}{\partial t} \approx n_{0} \int_{0}^{\infty} n_{0} \mathbf{C} P\left(n_{0} \mathbf{c}, t\right) d \mathbf{c}=n_{0} \int_{0}^{\infty} \mathbf{c} P^{c}(\mathbf{c}, t) d \mathbf{c}=n_{0} \frac{\partial\langle\mathbf{C}\rangle}{\partial t} nntP(n,t)n00n0CP(n0c,t)dc=n00cPc(c,t)dc=n0tC
所以
∂ ⟨ C ⟩ ∂ t = ∑ i = 1 m r s i ∫ 0 ∞ ⟨ M i C [ t − t ′ ; C ( t ′ ) ] } d t ′ \frac{\partial\langle\mathbf{C}\rangle}{\partial t}=\sum_{i=1}^{m_{r}} \mathbf{s}_{i} \int_{0}^{\infty}\left\langle M_{i}^{C}\left[t-t^{\prime} ; \mathbf{C}\left(t^{\prime}\right)\right]\right\} d t^{\prime} tC=i=1mrsi0MiC[tt;C(t)]}dt
其中 M i C [ t ; C ( t ) ] = M i [ t , n 0 C ( t ) ] / n 0 M_{i}^{C}[t ; \mathbf{C}(t)]=M_{i}\left[t, n_{0} \mathbf{C}(t)\right] / n_{0} MiC[t;C(t)]=Mi[t,n0C(t)]/n0

最后一行就是推广的化学速率方程

化学主方程的例子

考虑指数分布的内部反应等待时间,且无全局延迟。假设 p i ( t ) = κ i e − K t p_{i}(t)=\kappa_{i} e^{-K t} pi(t)=κieKt κ i \kappa_{i} κi为反应速率,则 ϕ i T = h i κ i exp ⁡ ( − ∑ l κ l h l t ) \phi_{i}^{T}=h_{i} \kappa_{i} \exp \left(-\sum_{l} \kappa_{l} h_{l} t\right) ϕiT=hiκiexp(lκlhlt),从而 M i = h i κ i δ ( t ) M_{i}=h_{i} \kappa_{i} \delta(t) Mi=hiκiδ(t)
此时,推广的化学主方程则退化为化学主方程:
∂ P ( n , t ) ∂ t = ∑ i = 1 m ∫ 0 t ( ∏ j = 1 m i E j − 5 j − 1 ) P ( n , t ′ ) h i κ i δ ( t − t ′ ) d t ′ = ∑ i = 1 m + h i κ i ( ∏ j = 1 m i E j − s i − 1 ) P ( n , t ) \begin{aligned} \frac{\partial P(\mathbf{n}, t)}{\partial t} &=\sum_{i=1}^{m} \int_{0}^{t}\left(\prod_{j=1}^{m_{i}} \mathrm{E}_{j}^{-5_{j}}-1\right) P\left(\mathbf{n}, t^{\prime}\right) h_{i} \kappa_{i} \delta\left(t-t^{\prime}\right) d t^{\prime} \\ &=\sum_{i=1}^{m_{+}} h_{i} \kappa_{i}\left(\prod_{j=1}^{m_{i}} \mathrm{E}_{j}^{-s_{i}}-1\right) P(\mathbf{n}, t) \end{aligned} tP(n,t)=i=1m0t(j=1miEj5j1)P(n,t)hiκiδ(tt)dt=i=1m+hiκi(j=1miEjsi1)P(n,t)
而对应的化学速率方程,在 n j n_{j} nj很大的情况下可以利用近似
h i ( n ) = ∏ j n j ! r j ! ( n j − r i j ) ! ≈ ∏ j n j n j r j ! h_{i}(\mathbf{n})=\prod_{j} \frac{n_{j} !}{r_{j} !\left(n_{j}-r_{i j}\right) !} \approx \prod_{j} \frac{n_{j}^{n_{j}}}{r_{j} !} hi(n)=jrj!(njrij)!nj!jrj!njnj ⟨ C j r j ⟩ ≈ ⟨ C j ⟩ η \left\langle C_{j}^{r_{j}}\right\rangle \approx\left\langle C_{j}\right\rangle^{\eta} CjrjCjη,最终可以求得的结果是:
∂ ⟨ C ⟩ ∂ t = ∑ i = 1 m s i κ i c ∏ j = 1 m ⟨ C j ⟩ ′ \frac{\partial\langle\mathbf{C}\rangle}{\partial t}=\sum_{i=1}^{m} s_{i} \kappa_{i}^{c} \prod_{j=1}^{m}\left\langle C_{j}\right\rangle^{\prime} tC=i=1msiκicj=1mCj
其中 κ i c = n 0 α i − 1 κ i / ∏ j = 1 m s r i j \kappa_{i}^{c}=n_{0}^{\alpha_{i}-1} \kappa_{i} / \prod_{j=1}^{m_{s}} r_{i j} κic=n0αi1κi/j=1msrij α i = ∑ j = 1 m s r j ˉ \alpha_{i}=\sum_{j=1}^{m_{s}} r_{\bar{j}} αi=j=1msrjˉ
至此,CTRW的化学主方程推导基本完成了。

CTRW模型在反常扩散中的研究与应用较多,结合分形动力学方程,如分数阶福克-普朗克方程,可以很好的描述复杂动力学系统中的反常输运现象。所以这方面的研究也不少。不过相对而言最近也没了太多的深入,可能是因为大家都去搞MLDL了吧(笑)。

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