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cg_zl [2014/08/21 17:42]
zl25
cg_zl [2014/08/21 17:45]
zl25
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 ===== Formulation of MZ-guided Markovian/​Non-Markovian DPD ===== ===== Formulation of MZ-guided Markovian/​Non-Markovian DPD =====
 The equation of motion (EOM) of the coarse-grained (CG) particles obtained from the Mori-Zwanzig projection is given by The equation of motion (EOM) of the coarse-grained (CG) particles obtained from the Mori-Zwanzig projection is given by
-  * Equation 1:+
 \begin{eqnarray}\label{equ:​EoM} \begin{eqnarray}\label{equ:​EoM}
 \frac{d}{dt}\mathbf{P}_I &=& \frac{1}{\beta} \frac{\partial}{\partial \mathbf{R}_I} {\rm{ln}} \omega(\mathbf{R}) \\ \nonumber \frac{d}{dt}\mathbf{P}_I &=& \frac{1}{\beta} \frac{\partial}{\partial \mathbf{R}_I} {\rm{ln}} \omega(\mathbf{R}) \\ \nonumber
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 where $\beta = 1/k_BT$ with $T$ the thermodynamic temperature and $k_B$ the Boltzmann constant, $\mathbf{R}=\{\mathbf{R}_1,​\mathbf{R}_2,​\cdots,​\mathbf{R}_K\}$ is a phase point in the CG phase space, and $\omega (\mathbf{R})$ is defined as a normalized partition function of all the microscopic configurations at phase point $\mathbf{R}$ given by where $\beta = 1/k_BT$ with $T$ the thermodynamic temperature and $k_B$ the Boltzmann constant, $\mathbf{R}=\{\mathbf{R}_1,​\mathbf{R}_2,​\cdots,​\mathbf{R}_K\}$ is a phase point in the CG phase space, and $\omega (\mathbf{R})$ is defined as a normalized partition function of all the microscopic configurations at phase point $\mathbf{R}$ given by
  
-  * Equation 2: 
 \begin{equation}\label{equ:​omega} \begin{equation}\label{equ:​omega}
   \omega (\mathbf{R})=\frac{\int d^N \mathbf{\hat r}\delta(\mathbf{\hat R}-\mathbf{R})e^{-\beta U} } {\int d^N \mathbf{\hat r} e^{-\beta U} } \ ,   \omega (\mathbf{R})=\frac{\int d^N \mathbf{\hat r}\delta(\mathbf{\hat R}-\mathbf{R})e^{-\beta U} } {\int d^N \mathbf{\hat r} e^{-\beta U} } \ ,
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   \frac{1}{\beta} \frac{\partial}{\partial \mathbf{R}_I} {\rm{ln}} \omega(\mathbf{R}) = \langle \mathbf{F}_I \rangle \approx \sum_{J\neq I}\langle \mathbf{F}_{IJ} \rangle = \sum_{J\neq I}{F_{IJ}^C}(R_{IJ})\mathbf{e}_{IJ}   \frac{1}{\beta} \frac{\partial}{\partial \mathbf{R}_I} {\rm{ln}} \omega(\mathbf{R}) = \langle \mathbf{F}_I \rangle \approx \sum_{J\neq I}\langle \mathbf{F}_{IJ} \rangle = \sum_{J\neq I}{F_{IJ}^C}(R_{IJ})\mathbf{e}_{IJ}
 \end{equation} \end{equation}
-\noindent+
 where $\mathbf{e}_{IJ}$ is the unit vector from CG particle $J$ to $I$ given by $\mathbf{e}_{I J}=(\mathbf{R}_I-\mathbf{R}_J)/​R_{I J}$ with $R_{I J}=|\mathbf{R}_I-\mathbf{R}_J|$. The rotational symmetry of the CG pairs about the $\mathbf{e}_{IJ}$ suggests that, on average, $\mathbf{F}_{IJ}$ has no preference on the plane perpendicular to $\mathbf{e}_{IJ}$ and remains only nonzero component along $\mathbf{e}_{IJ}$. Here, $F_{IJ}^C(R_{I J})$ represents the magnitude of conservative force $\mathbf{F}_{IJ}^C$,​ which is time independent but distance dependent. where $\mathbf{e}_{IJ}$ is the unit vector from CG particle $J$ to $I$ given by $\mathbf{e}_{I J}=(\mathbf{R}_I-\mathbf{R}_J)/​R_{I J}$ with $R_{I J}=|\mathbf{R}_I-\mathbf{R}_J|$. The rotational symmetry of the CG pairs about the $\mathbf{e}_{IJ}$ suggests that, on average, $\mathbf{F}_{IJ}$ has no preference on the plane perpendicular to $\mathbf{e}_{IJ}$ and remains only nonzero component along $\mathbf{e}_{IJ}$. Here, $F_{IJ}^C(R_{I J})$ represents the magnitude of conservative force $\mathbf{F}_{IJ}^C$,​ which is time independent but distance dependent.
  
-\par 
 Similarly, the fluctuating force defined as the deviation from the mean force is also decomposed into pairwise forces Similarly, the fluctuating force defined as the deviation from the mean force is also decomposed into pairwise forces
 \begin{eqnarray}\label{equ:​FR} \begin{eqnarray}\label{equ:​FR}
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  ​\delta\mathbf{F}_{IJ} &=& \mathbf{F}_{IJ}- \langle \mathbf{F}_{IJ} \rangle = \mathbf{F}_{IJ} -{F_{IJ}^C}(R_{IJ})\mathbf{e}_{IJ}  ​\delta\mathbf{F}_{IJ} &=& \mathbf{F}_{IJ}- \langle \mathbf{F}_{IJ} \rangle = \mathbf{F}_{IJ} -{F_{IJ}^C}(R_{IJ})\mathbf{e}_{IJ}
 \end{eqnarray} \end{eqnarray}
-\noindent+
 where $\mathbf{F}_{IJ}$ is the instantaneous force exerted by cluster $J$ on cluster $I$, and $\langle \mathbf{F}_{IJ}\rangle$ is the ensemble average of $\mathbf{F}_{IJ}$ obtained by Eq. (\ref{equ:​FC}). where $\mathbf{F}_{IJ}$ is the instantaneous force exerted by cluster $J$ on cluster $I$, and $\langle \mathbf{F}_{IJ}\rangle$ is the ensemble average of $\mathbf{F}_{IJ}$ obtained by Eq. (\ref{equ:​FC}).
  
-\par 
 Now we consider the memory kernel in Eq. (\ref{equ:​EoM}). Based on the second approximation,​ the correlation of fluctuating forces between different pairs is ignored. Thus, we have Now we consider the memory kernel in Eq. (\ref{equ:​EoM}). Based on the second approximation,​ the correlation of fluctuating forces between different pairs is ignored. Thus, we have
 \begin{eqnarray}\label{equ:​no_corelation} \begin{eqnarray}\label{equ:​no_corelation}
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   &​=&​\left \langle [\delta\mathbf{F}_{IJ}(t-s)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle \mathbf{V}_{IJ}(s)   &​=&​\left \langle [\delta\mathbf{F}_{IJ}(t-s)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle \mathbf{V}_{IJ}(s)
 \end{eqnarray} \end{eqnarray}
-\noindent + 
-where $\mathbf{V}_{IJ}=\mathbf{V}_{J}-\mathbf{V}_{J}$ is the relative velocity of CG particle $I$ to $J$. Moreover, {\color{red}Third approximation:​} we assume that the memory on time is finite, e.g. history length $N\cdot\Delta t$ where $\Delta t$ is the time step of DPD simulations. Therefore, the time correlation between the fluctuating forces is zero when the time interval is larger than $N\Delta t$+where $\mathbf{V}_{IJ}=\mathbf{V}_{J}-\mathbf{V}_{J}$ is the relative velocity of CG particle $I$ to $J$. Moreover, ​${\color{red}{Thirdapproximation:​}}$ we assume that the memory on time is finite, e.g. history length $N\cdot\Delta t$ where $\Delta t$ is the time step of DPD simulations. Therefore, the time correlation between the fluctuating forces is zero when the time interval is larger than $N\Delta t$
 \begin{equation}\label{equ:​no_history} \begin{equation}\label{equ:​no_history}
 \left\langle[\delta\mathbf{F}_I(t)] [\delta\mathbf{F}_X(0)]^T \right \rangle|_{t>​N\Delta t} = 0 \left\langle[\delta\mathbf{F}_I(t)] [\delta\mathbf{F}_X(0)]^T \right \rangle|_{t>​N\Delta t} = 0
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   &=& -{\beta\cdot \Delta t}\sum_{J\neq I}\sum_{n=0}^{N}\alpha_n \left \langle [\delta\mathbf{F}_{IJ}(n\Delta t)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle \mathbf{V}_{IJ}(t-n\Delta t)   &=& -{\beta\cdot \Delta t}\sum_{J\neq I}\sum_{n=0}^{N}\alpha_n \left \langle [\delta\mathbf{F}_{IJ}(n\Delta t)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle \mathbf{V}_{IJ}(t-n\Delta t)
 \end{eqnarray} \end{eqnarray}
-\par+
 We define the friction matrix ${\Gamma}_{IJ,​n}$ as We define the friction matrix ${\Gamma}_{IJ,​n}$ as
 \begin{equation}\label{equ:​Gamma} \begin{equation}\label{equ:​Gamma}
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 where $\delta\mathbf{F}^{\parallel}_{IJ}$ is the component along vector $\mathbf{e}_{IJ}$ and $\delta\mathbf{F}^{\perp}_{IJ}$ the perpendicular part whose modulus is equally distributed on directions $\perp_1$ and $\perp_2$. where $\delta\mathbf{F}^{\parallel}_{IJ}$ is the component along vector $\mathbf{e}_{IJ}$ and $\delta\mathbf{F}^{\perp}_{IJ}$ the perpendicular part whose modulus is equally distributed on directions $\perp_1$ and $\perp_2$.
  
-**Remark:** The memory term given by Eq. (\ref{equ:​FD}) can be further simplified with a {\color{red}Markovian assumption} that the memory of fluctuating force in time is short enough to be approximated by a Dirac delta function+**Remark:** The memory term given by Eq. (\ref{equ:​FD}) can be further simplified with a ${\color{red}{Markovianassumption}}$ that the memory of fluctuating force in time is short enough to be approximated by a Dirac delta function
  
 \begin{eqnarray} \begin{eqnarray}

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