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\begin{document}

\title{ColourNoise_Formula}


\author[ ]{Mengkai Feng}

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\vspace{-1em}


\date{}

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\begin{document}

\section{MCT for colour noise}

Mode coupling theory derivation of active particles, with colour noise
Smoluchowski operator, using Kawasaki's irreducible projection method.

\[
\frac{\partial}{\partial t}C_{A}(t)+\omega_{A}C_{A}(t)+\int_{0}^{t}du\tilde{M}_{A}^{irr}\left(t-u\right)\frac{\partial}{\partial u}C_{A}(u)=0
\]
\[
\omega_{A}=-\frac{\left\langle A^{*}\Omega A\right\rangle }{C_{A}(0)}
\]
\[
\tilde{M}_{A}^{irr}\left(t\right)=-\frac{\left\langle A^{*}\Omega\mathcal{Q}e^{\mathcal{Q}\Omega^{irr}\mathcal{Q}t}\mathcal{Q}\Omega A\right\rangle }{\left\langle A^{*}\Omega A\right\rangle }
\]
\[
\left\langle \cdots\right\rangle =\int d\Gamma\cdots P_{s}(\Gamma)
\]





\subsection{Collective Behaviour}

\[
A(t)=\delta\rho_{q}(t)=\sum_{j=1}^{N}e^{-i\mathbf{q}\cdot\mathbf{r}_{j}(t)}
\]
\[
C_{A}(0)=NS(q)
\]
\[
\Omega=\sum_{j=1}^{N}\nabla_{j}\cdot D_{j}(r^{N})\left(\nabla_{j}-\beta\mathbf{F}_{j}^{{\rm eff}}\right)
\]
\[
\Omega^{\dagger}=\sum_{j=1}^{N}\left(\nabla_{j}+\beta\mathbf{F}_{j}^{{\rm eff}}\right)D_{j}\cdot\nabla_{j}
\]
\[
\Omega^{\dagger}A=\sum_{j}\left(\nabla_{j}+\beta\mathbf{F}_{j}^{{\rm eff}}\right)D_{j}\cdot\left(-i\mathbf{q}\right)e^{-i\mathbf{q}\cdot\mathbf{r}_{j}}
\]
\begin{eqnarray*}
\left\langle A^{*}\Omega A\right\rangle  & = & \left\langle A^{*}\left(\Omega^{\dagger}A\right)\right\rangle \\
 & = & \left\langle w_{1}+w_{2}+w_{3}\right\rangle 
\end{eqnarray*}
\[
w_{1}=\sum_{i,j}\left(\nabla_{j}D_{j}\right)\cdot\left(-i\mathbf{q}\right)e^{-i\mathbf{q}\cdot\left(\mathbf{r}_{j}-\mathbf{r}_{i}\right)}
\]
\[
w_{2}=\sum_{i,j}-D_{j}q^{2}e^{-i\mathbf{q}\cdot\left(\mathbf{r}_{j}-\mathbf{r}_{i}\right)}
\]
\[
w_{3}=\sum_{i,j}\beta\mathbf{F}_{j}^{{\rm eff}}D_{j}\cdot\left(-i\mathbf{q}\right)e^{-i\mathbf{q}\cdot\left(\mathbf{r}_{j}-\mathbf{r}_{i}\right)}
\]
Considering $\partial_{t}P_{s}=\Omega P_{s}=0$, so $\beta\mathbf{F}_{i}^{{\rm eff}}=\nabla_{i}\ln P_{s}$,
then we have Yvon theorem:
\[
\left\langle B\beta\mathbf{F}_{j}^{{\rm eff}}\right\rangle =-\left\langle \nabla_{j}B\right\rangle 
\]
Then
\begin{eqnarray*}
\left\langle w_{3}\right\rangle  & = & \left\langle \sum_{i,j}\nabla_{j}\cdot\left[i\mathbf{q}D_{j}e^{-i\mathbf{q}\cdot\left(\mathbf{r}_{j}-\mathbf{r}_{i}\right)}\right]\right\rangle \\
 & = & \sum_{i,j}\left(\left\langle i\mathbf{q}\cdot\left(\nabla_{j}D_{j}\right)e^{-i\mathbf{q}\cdot\left(\mathbf{r}_{j}-\mathbf{r}_{i}\right)}\right\rangle +\left\langle q^{2}D_{j}e^{-i\mathbf{q}\cdot\left(\mathbf{r}_{j}-\mathbf{r}_{i}\right)}\right\rangle +\delta_{ij}\left\langle -q^{2}D_{j}e^{-i\mathbf{q}\cdot\left(\mathbf{r}_{j}-\mathbf{r}_{i}\right)}\right\rangle \right)\\
 & = & -\left\langle w_{1}+w_{2}\right\rangle +\left\langle -q^{2}\sum_{j}D_{j}\right\rangle 
\end{eqnarray*}
\[
\left\langle A^{*}\Omega A\right\rangle =-q^{2}\sum_{j}\left\langle D_{j}\right\rangle 
\]
\[
\omega_{A}=\frac{q^{2}\sum_{j}\left\langle D_{j}\right\rangle }{NS(q)}
\]



If $D_{j}=D_{0}=const$ like equilibrium situation or equivalent temperature
framwork, $\left\langle A^{*}\Omega A\right\rangle =-q^{2}ND_{0}$.
However, we have $D_{j}=D_{t}+\frac{D_{a}}{1-\tau\nabla_{j}\cdot\beta\mathbf{F}_{j}}$$=D_{t}+\frac{D_{a}}{1+d^{2}\beta\nabla_{j}^{2}U_{j}/6}$.
This should be determined by simulations. For convenience, we define
$\alpha ND_{0}=\sum_{j}\left\langle D_{j}\right\rangle $, where $D_{0}=D_{a}+D_{t}$.
 
\begin{eqnarray*}
\left\langle A^{*}\Omega\mathcal{Q}e^{\mathcal{Q}\Omega^{irr}\mathcal{Q}t}\mathcal{Q}\Omega A\right\rangle  & = & \left\langle \left(\Omega^{\dagger}A\right)^{*}\mathcal{Q}e^{\mathcal{Q}\Omega^{irr}\mathcal{Q}t}\mathcal{Q}\left(\Omega^{\dagger}A\right)\right\rangle \\
 & = & \left\langle R_{q}^{*}e^{\mathcal{Q}\Omega^{irr}\mathcal{Q}t}R_{q}\right\rangle 
\end{eqnarray*}
\begin{eqnarray*}
R_{q} & = & \mathcal{Q}\left(\Omega^{\dagger}A\right)=(\Omega^{\dagger}A)-\mathcal{P}(\Omega^{\dagger}A)\\
 & = & (\Omega^{\dagger}A)-\frac{\left\langle A^{*}\left(\Omega^{\dagger}A\right)\right\rangle }{\left\langle A^{*}A\right\rangle }A\\
 & = & (\Omega^{\dagger}A)+\frac{q^{2}\sum_{j}\left\langle D_{j}\right\rangle }{NS(q)}A
\end{eqnarray*}
Mode-coupling Approximation: 
\begin{eqnarray*}
\left\langle R_{q}^{*}e^{\mathcal{Q}\Omega^{irr}\mathcal{Q}t}R_{q}\right\rangle  & \approx & \sum\left|\left\langle \left(AA\right)^{*}R_{q}\right\rangle \right|^{2}\left\langle \left(AA\right)^{*}AA\right\rangle ^{-2}\left\langle \left(AA\right)^{*}e^{\mathcal{Q}\Omega^{irr}\mathcal{Q}t}AA\right\rangle \\
 & \approx & \sum\left|V\left(AA,R_{q}\right)\right|^{2}\left\langle A^{*}e^{\Omega t}A\right\rangle \left\langle A^{*}e^{\Omega t}A\right\rangle 
\end{eqnarray*}
where
\[
V\left(AA,R_{q}\right)=\left\langle \left(AA\right)^{*}R_{q}\right\rangle \left\langle \left(AA\right)^{*}AA\right\rangle ^{-1}
\]
\begin{eqnarray*}
\left\langle \left(AA\right)^{*}AA\right\rangle  & = & \left\langle \delta\rho_{\mathbf{p}}^{*}\delta\rho_{\mathbf{k}}^{*}\delta\rho_{\mathbf{p}}\delta\rho_{\mathbf{k}}\right\rangle \\
 & \approx & \left\langle \delta\rho_{\mathbf{p}}^{*}\delta\rho_{\mathbf{k}}\right\rangle \left\langle \delta\rho_{\mathbf{k}}^{*}\delta\rho_{\mathbf{p}}\right\rangle +\left\langle \delta\rho_{\mathbf{p}}^{*}\delta\rho_{\mathbf{p}}\right\rangle \left\langle \delta\rho_{\mathbf{k}}^{*}\delta\rho_{\mathbf{k}}\right\rangle \\
 & = & N^{2}S(p)S(k)
\end{eqnarray*}
\[
\left\langle \left(AA\right)^{*}R_{q}\right\rangle =\left\langle \left(AA\right)^{*}(\Omega^{\dagger}A)\right\rangle +\frac{q^{2}\sum_{j}\left\langle D_{j}\right\rangle }{NS(q)}\left\langle \left(AA\right)^{*}A\right\rangle =\left\langle R_{1}+R_{2}+R_{3}+R_{4}\right\rangle 
\]
\[
\left\langle R_{1}\right\rangle =\left\langle \left(\sum_{i,l}e^{i\mathbf{k}\cdot\mathbf{r}_{i}+i\mathbf{p}\cdot\mathbf{r}_{l}}\right)\sum_{j}\left(\nabla_{j}D_{j}\right)\cdot\left(-i\mathbf{q}\right)e^{-i\mathbf{q}\cdot\mathbf{r}_{j}}\right\rangle 
\]
\[
\left\langle R_{2}\right\rangle =\left\langle \left(\sum_{i,l}e^{i\mathbf{k}\cdot\mathbf{r}_{i}+i\mathbf{p}\cdot\mathbf{r}_{l}}\right)\sum_{j}\left(-D_{j}q^{2}e^{-i\mathbf{q}\cdot\mathbf{r}_{j}}\right)\right\rangle 
\]
\begin{eqnarray*}
\left\langle R_{3}\right\rangle  & = & \left\langle \left(\sum_{i,l}e^{i\mathbf{k}\cdot\mathbf{r}_{i}+i\mathbf{p}\cdot\mathbf{r}_{l}}\right)\sum_{j}\beta\mathbf{F}_{j}^{{\rm eff}}D_{j}\cdot\left(-i\mathbf{q}\right)e^{-i\mathbf{q}\cdot\mathbf{r}_{j}}\right\rangle \\
 & = & \left\langle \sum_{i,j,l}\left(i\mathbf{q}\right)\cdot\nabla_{j}\left[D_{j}e^{-i\mathbf{q}\cdot\mathbf{r}_{j}+i\mathbf{k}\cdot\mathbf{r}_{i}+i\mathbf{p}\cdot\mathbf{r}_{l}}\right]\right\rangle \\
 & = & -\left\langle R_{1}\right\rangle -\left\langle R_{2}\right\rangle +\left\langle R_{31}+R_{32}\right\rangle 
\end{eqnarray*}



\begin{eqnarray*}
\left\langle R_{31}\right\rangle  & = & \delta_{ij}\left\langle \sum_{i,j,l}\left(-\mathbf{q}\cdot\mathbf{k}\right)D_{j}e^{-i\mathbf{q}\cdot\mathbf{r}_{j}+i\mathbf{k}\cdot\mathbf{r}_{i}+i\mathbf{p}\cdot\mathbf{r}_{l}}\right\rangle \\
 & = & \left\langle \sum_{j,l}\left(-\mathbf{q}\cdot\mathbf{k}\right)D_{j}e^{-i\left(\mathbf{q}-\mathbf{k}\right)\cdot\mathbf{r}_{j}+i\mathbf{p}\cdot\mathbf{r}_{l}}\right\rangle 
\end{eqnarray*}
\begin{eqnarray*}
\left\langle R_{32}\right\rangle  & = & \delta_{il}\left\langle \sum_{i,j,l}\left(-\mathbf{q}\cdot\mathbf{p}\right)D_{j}e^{-i\mathbf{q}\cdot\mathbf{r}_{j}+i\mathbf{k}\cdot\mathbf{r}_{i}+i\mathbf{p}\cdot\mathbf{r}_{l}}\right\rangle \\
 & = & \left\langle \sum_{i,j}\left(-\mathbf{q}\cdot\mathbf{p}\right)D_{j}e^{-i\left(\mathbf{q}-\mathbf{p}\right)\cdot\mathbf{r}_{j}+i\mathbf{k}\cdot\mathbf{r}_{i}}\right\rangle 
\end{eqnarray*}
If $D_{j}$ is not relevant to spatial variables like our works before,
we have $\left\langle R_{31}\right\rangle =-\left(\mathbf{q}\cdot\mathbf{k}\right)ND\delta_{\mathbf{q}-\mathbf{k},\mathbf{p}}S(p)$.
But here we can't take $D_{j}$ out of the bracket, we have to use
simulation method. For convenience, here we define $D_{0}NS_{2}(p)=\left\langle \sum_{i,j}D_{j}e^{-i\mathbf{p}\cdot\mathbf{r}_{j}+i\mathbf{p}\cdot\mathbf{r}_{i}}\right\rangle $,
which is close to $D_{0}NS(p)$.  
\[
\left\langle R_{4}\right\rangle =\frac{q^{2}\sum_{j}\left\langle D_{j}\right\rangle }{NS(q)}\left\langle \delta\rho_{\mathbf{p}}^{*}\delta\rho_{\mathbf{k}}^{*}\delta\rho_{\mathbf{q}}\right\rangle \approx\delta_{\mathbf{k}+\mathbf{p},\mathbf{q}}q^{2}\sum_{j}\left\langle D_{j}\right\rangle S(p)S(k)
\]
\begin{eqnarray*}
\left\langle \left(AA\right)^{*}R_{q}\right\rangle  & = & -\delta_{\mathbf{q}-\mathbf{k},\mathbf{p}}\left(\mathbf{q}\cdot\mathbf{k}\right)D_{0}NS_{2}(p)-\delta_{\mathbf{q}-\mathbf{p},\mathbf{k}}\left(\mathbf{q}\cdot\mathbf{p}\right)D_{0}NS_{2}(k)\\
 &  & +\delta_{\mathbf{k}+\mathbf{p},\mathbf{q}}q^{2}\alpha ND_{0}S(p)S(k)
\end{eqnarray*}
\begin{eqnarray*}
V\left(AA,R_{q}\right) & = & \left\langle \left(AA\right)^{*}R_{q}\right\rangle \left\langle \left(AA\right)^{*}AA\right\rangle ^{-1}\\
 & = & -\delta_{\mathbf{k}+\mathbf{p},\mathbf{q}}\left\{ \left(\mathbf{q}\cdot\mathbf{k}\right)\frac{D_{0}S_{2}(p)}{NS(p)S(k)}+\left(\mathbf{q}\cdot\mathbf{p}\right)\frac{D_{0}S_{2}(k)}{NS(p)S(k)}-\frac{q^{2}\alpha D_{0}}{N}\right\} \\
 & = & -\delta_{\mathbf{k}+\mathbf{p},\mathbf{q}}\frac{D_{0}}{N}\left\{ \left(\mathbf{q}\cdot\mathbf{k}\right)\left[\frac{S_{2}(p)}{S(p)S(k)}-\alpha\right]+\left(\mathbf{q}\cdot\mathbf{p}\right)\left[\frac{S_{2}(k)}{S(p)S(k)}-\alpha\right]\right\} 
\end{eqnarray*}
Still we can define a pseudo ``Direct correlation function'' $C_{2}(p;k)=\frac{\delta_{\mathbf{k}+\mathbf{p},\mathbf{q}}}{\rho}\left[\alpha-\frac{S_{2}(p)}{S(p)S(k)}\right]$
. Then we have 
\begin{eqnarray*}
V\left(AA,R_{q}\right) & = & V\left(q,k\right)\\
 & = & \frac{D_{0}}{V}\left[\left(\mathbf{q}\cdot\mathbf{k}\right)C_{2}\left(p;k\right)+\left(\mathbf{q}\cdot\mathbf{p}\right)C_{2}(k;p)\right]
\end{eqnarray*}
\[
\left\langle R_{q}^{*}e^{\mathcal{Q}\Omega^{irr}\mathcal{Q}t}R_{q}\right\rangle \approx\sum_{\mathbf{k}}\frac{D_{0}^{2}}{V^{2}}\left[\left(\mathbf{q}\cdot\mathbf{k}\right)C_{2}\left(p;k\right)+\left(\mathbf{q}\cdot\mathbf{p}\right)C_{2}(k;p)\right]^{2}C_{A}(k,t)C_{A}(p,t)
\]
\begin{eqnarray*}
\tilde{M}^{irr}\left(q;t\right) & \approx & \frac{D_{0}}{2q^{2}NV^{2}\alpha}\sum_{\mathbf{k}}\left[\left(\mathbf{q}\cdot\mathbf{k}\right)C_{2}\left(p;k\right)+\left(\mathbf{q}\cdot\mathbf{p}\right)C_{2}(k;p)\right]^{2}C_{A}(k,t)C_{A}(p,t)\\
 & = & \frac{D_{0}}{16\pi^{3}NV\alpha}\int d^{3}\mathbf{k}\left[\left(\hat{\mathbf{q}}\cdot\mathbf{k}\right)C_{2}\left(p;k\right)+\left(\hat{\mathbf{q}}\cdot\mathbf{p}\right)C_{2}(k;p)\right]^{2}C_{A}(k,t)C_{A}(p,t)\\
 & = & \frac{D_{0}}{8\pi^{2}NV\alpha}\int_{0}^{\pi}\sin\theta d\theta\int_{0}^{\infty}k^{2}dk\left[\left(\hat{\mathbf{q}}\cdot\mathbf{k}\right)C_{2}\left(p;k\right)+\left(\hat{\mathbf{q}}\cdot\mathbf{p}\right)C_{2}(k;p)\right]^{2}C_{A}(k,t)C_{A}(p,t)\\
 & = & \frac{\rho D_{0}}{8\pi^{2}\alpha}\int_{0}^{\infty}\int_{\left|q-k\right|}^{q+k}dp\frac{pk}{q}dk\left[\frac{k^{2}+q^{2}-p^{2}}{2q}C_{2}\left(p;k\right)+\frac{p^{2}+q^{2}-k^{2}}{2q}C_{2}(k;p)\right]^{2}S(k)S(p)\phi_{k}(t)\phi_{p}(t)\\
 & = & \frac{\rho D_{0}}{32\pi^{2}\alpha}\int_{0}^{\infty}\int_{\left|q-k\right|}^{q+k}dp\frac{pk}{q^{3}}dk\left[\left(k^{2}+q^{2}-p^{2}\right)C_{2}\left(p;k\right)+\left(p^{2}+q^{2}-k^{2}\right)C_{2}(k;p)\right]^{2}S(k)S(p)\phi_{k}(t)\phi_{p}(t)
\end{eqnarray*}



\subsection{Tagged Particle}

\[
A=\delta\rho_{q}^{s}(t)=e^{-i\mathbf{q}\cdot\mathbf{r}_{s}(t)}
\]

\[
C(t)=F_{s}(q,t)=\left\langle \delta\rho_{q}^{s}(t)\delta\rho_{-q}^{s}(0)\right\rangle 
\]
\[
F_{s}(q,0)=1
\]
\[
\Omega^{\dagger}A=\left(\nabla_{s}+\beta\mathbf{F}_{s}^{{\rm eff}}\right)D_{s}\cdot\left(-i\mathbf{q}\right)e^{-i\mathbf{q}\cdot\mathbf{r}_{s}}
\]
\[
\left\langle A^{*}\Omega A\right\rangle =\left\langle A^{*}\left(\Omega^{\dagger}A\right)\right\rangle =-q^{2}\left\langle D_{s}\right\rangle 
\]
\[
\omega_{A}=q^{2}\left\langle D_{s}\right\rangle 
\]
\begin{eqnarray*}
R_{q} & = & \mathcal{Q}\left(\Omega^{\dagger}A\right)=(\Omega^{\dagger}A)-\mathcal{P}(\Omega^{\dagger}A)\\
 & = & (\Omega^{\dagger}A)-\frac{\left\langle A^{*}\left(\Omega^{\dagger}A\right)\right\rangle }{\left\langle A^{*}A\right\rangle }A\\
 & = & (\Omega^{\dagger}A)+q^{2}\left\langle D_{s}\right\rangle A
\end{eqnarray*}
\[
AA=\delta\rho_{\mathbf{p}}^{s}\delta\rho_{\mathbf{k}}
\]
\[
\left\langle \left(AA\right)^{*}AA\right\rangle =\left\langle \delta\rho_{\mathbf{p}}^{s*}\delta\rho_{\mathbf{k}}^{*}\delta\rho_{\mathbf{p}}^{s}\delta\rho_{\mathbf{k}}\right\rangle =NS(k)
\]
\[
\left\langle \left(AA\right)^{*}R_{q}\right\rangle =\left\langle \left(AA\right)^{*}(\Omega^{\dagger}A)\right\rangle +q^{2}\left\langle D_{s}\right\rangle \left\langle \left(AA\right)^{*}A\right\rangle 
\]


\section{Athermal situation}

Ornstein-Uhlenbeck process:
\begin{eqnarray*}
\dot{\mathbf{r}}_{i} & = & \xi_{0}^{-1}\left[\mathbf{F}_{i}+\mathbf{f}_{i}\right]\\
\dot{\mathbf{f}}_{i} & = & -\tau_{p}^{-1}\mathbf{f}_{i}+\bf{\eta}_{i}
\end{eqnarray*}
Averaging over the angular degrees of freedom generates a coarse-grained
equation:
\[
\dot{\mathbf{r}}_{i}=\xi_{0}^{-1}\mathbf{F}_{i}+\bm{\chi}_{i}
\]
than we have the Smoluchowski operator: 
\begin{eqnarray*}
\Omega & = & -\sum_{i=1}^{N}\nabla_{i}\cdot\xi_{0}^{-1}\mathbf{F}_{i}+\frac{D_{f}\tau_{p}^{2}}{\xi_{0}^{2}}\sum_{i=1}^{N}\nabla_{i}^{2}\frac{1}{1-\tau_{p}\xi_{0}^{-1}\nabla_{i}\cdot\mathbf{F}_{i}}\\
 & = & \sum_{i=1}^{N}\nabla_{i}\cdot D_{i}\left[\nabla_{i}-\frac{1}{\xi_{0}D_{i}}\mathbf{F}_{i}^{{\rm eff}}\right]
\end{eqnarray*}
where $D_{i}\left(\mathbf{r}^{N}\right)=\frac{D_{f}\tau_{p}^{2}}{\xi_{0}^{2}}\left(1-\tau_{p}\xi_{0}^{-1}\nabla_{i}\cdot\mathbf{F}_{i}\right)^{-1}$,
$\mathbf{F}_{i}^{{\rm eff}}=\mathbf{F}_{i}-\frac{D_{f}\tau_{p}^{2}}{\xi_{0}}\nabla_{i}\frac{1}{1-\tau_{p}\xi_{0}^{-1}\nabla_{i}\cdot\mathbf{F}_{i}}=\mathbf{F}_{i}-\frac{1}{\xi_{0}}\nabla_{i}D_{i}$. 

In the steady state, we have $\Omega P_{ss}(\mathbf{r}^{N})=0$, then
$D_{t}\xi_{0}\nabla_{i}\ln P_{ss}(\mathbf{r}^{N})=\mathbf{F}_{i}^{{\rm eff}}$.
Comparing with Szamel's paper where got $\left\langle \mathbf{f}_{i}\right\rangle _{lss}=-\mathbf{F}_{i}$,
we can notice that $\left\langle \mathbf{f}_{i}\right\rangle _{lss}=-\left(\xi_{0}^{-1}\nabla_{i}D_{i}\left(\mathbf{r}^{N}\right)+D_{0}\xi_{0}\nabla_{i}\ln P_{ss}\left(\mathbf{r}^{N}\right)\right)$.
It must be said that in his paper, he defined $\mathbf{F}_{i}^{{\rm eff}}=\nabla_{i}\ln P_{ss}(\mathbf{r}^{N})$
which is confusing and ``wrong'' at dimension. I used this in my
derivations before and regret it now. 

Next, I put this Smoluchowski operator into the MCT framwork. The
procedure is same with last section, so we give the result directly.
\[
\frac{\partial}{\partial t}F(q,t)+\omega(q)F(q,t)+\int_{0}^{t}duM_{q}^{irr}\left(t-u\right)\frac{\partial}{\partial u}F(q,u)=0
\]
\[
\omega(q)=\frac{q^{2}\sum_{i}\left\langle D_{i}\right\rangle }{NS(q)}
\]
\[
M_{q}^{irr}\left(t\right)=-\frac{D_{0}}{N\alpha}\left[\left(\hat{\mathbf{q}}\cdot\mathbf{k}\right)C_{2}\left(k;q\right)+\left(\hat{\mathbf{q}}\cdot\mathbf{p}\right)C_{2}(p;q)\right]^{2}F(p,t)F(k,t)
\]
This outcome could compare with Szamel's result
\[
\partial_{t}^{2}F(q,t)+\tau_{p}^{-1}\partial_{t}F(q,t)+\frac{q^{2}\omega_{\shortparallel}(q)}{S(q)}F(q,t)+\int_{0}^{t}dt'M^{{\rm irr}}(q,t-t')\partial_{t'}F(q,t')=0
\]





\subsection{Contract}

In Szamel's paper, 
\[
\partial_{t}^{2}F(q,t)+\tau_{p}^{-1}\partial_{t}F(q,t)+\frac{q^{2}\omega_{\shortparallel}(q)}{S(q)}F(q,t)+\int_{0}^{t}dt'M^{{\rm irr}}(q,t-t')\partial_{t'}F(q,t')=0
\]
\[
\omega_{\shortparallel}\left(q\right)=\frac{1}{N\xi^{2}}\left\langle \left|\hat{{\bf q}}\cdot\sum_{i}\left({\bf f}_{i}+{\bf F}_{i}\right)e^{-i{\bf q}\cdot{\bf r}_{i}}\right|^{2}\right\rangle 
\]
\[
M^{irr}(q,t)=\frac{\rho\omega_{\shortparallel}\left(q\right)}{2}\int\frac{d{\bf k}}{\left(2\pi\right)^{3}}\left(\hat{{\bf q}}\cdot\left[{\bf k}\mathcal{C}\left(k\right)+{\bf p}\mathcal{C}\left(p\right)\right]\right)^{2}F\left(k,t\right)F\left(p,t\right)
\]
\[
\mathcal{C}(q)=\frac{1}{\rho}\left[1-\frac{\omega_{\shortparallel}\left(q\right)}{\omega_{\shortparallel}\left(\infty\right)S\left(q\right)}\right]
\]
\[
\omega_{\shortparallel}\left(\infty\right)=\frac{1}{3N\xi^{2}}\left\langle \sum_{i}\left({\bf f}_{i}+{\bf F}_{i}\right)^{2}\right\rangle =\lim_{q\rightarrow\infty}\omega_{\shortparallel}\left(q\right)
\]
Interestingly we find 
\[
\frac{\tau_{p}}{3N\xi^{2}}\left\langle \sum_{i}\left({\bf f}_{i}+{\bf F}_{i}\right)^{2}\right\rangle =\frac{\tau_{p}}{3N}\sum_{i}\left\langle \dot{{\bf r}}_{i}^{2}\right\rangle 
\]
\[
\frac{\partial}{\partial t}F_{q}(t)+\omega_{q}F_{q}(t)+\int_{0}^{t}du\tilde{M}^{irr}\left(q,t-u\right)\frac{\partial}{\partial u}F_{q}(u)=0
\]
\[
\omega_{q}=\frac{q^{2}\alpha D_{0}}{S(q)}
\]
\[
\tilde{M}^{irr}\left(q,t\right)=\frac{\rho\alpha D_{0}}{16\pi^{3}}\int d\mathbf{k}\left[\left(\hat{{\bf q}}\cdot\mathbf{k}\right)C_{2}\left(q,k\right)+\left(\hat{{\bf q}}\cdot\mathbf{p}\right)C_{2}\left(q,p\right)\right]^{2}F\left(k,t\right)F\left(p,t\right)
\]
\[
C_{2}\left(q,k\right)=\rho^{-1}\left[1-\frac{S_{2}\left(p\right)}{\alpha S\left(p\right)}S^{-1}\left(k\right)\right]
\]
Directly, we have corrosponding relationship: $\frac{q^{2}\omega_{\shortparallel}(q)\tau_{p}}{S(q)}\sim\frac{q^{2}\alpha D_{0}}{S(q)}$,
$\rho\omega_{\shortparallel}\left(q\right)\tau_{p}\sim\rho\alpha D_{0}$,
i. e. $\omega_{\shortparallel}(q)\tau_{p}\sim\alpha D_{0}=\bar{D}=\frac{D_{f}\tau_{p}^{2}}{\xi_{0}^{2}}\left\langle \left(1-\tau_{p}\xi_{0}^{-1}\nabla_{i}\cdot\mathbf{F}_{i}\right)^{-1}\right\rangle $. 

In the limit of $\tau_{p}\rightarrow0$, $\omega_{\shortparallel}\left(q\right)\rightarrow\omega_{\shortparallel}\left(\infty\right)$,
$\mathcal{C}\left(q\right)\rightarrow c(q)$. (Szamel's simulation
results), under this circumstance, logically we should have 
\[
=\frac{D_{f}\tau_{p}^{2}}{\xi_{0}^{2}}\left\langle \left(1-\tau_{p}\xi_{0}^{-1}\nabla_{i}\cdot\mathbf{F}_{i}\right)^{-1}\right\rangle 
\]

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