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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, ${\color{red}{Third\ approximation:}}$ 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 $N\Delta t$ |
\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 ${\bm \Gamma}_{IJ,n}$ as | + | We define the friction matrix ${\Gamma}_{IJ,n}$ as |
\begin{equation}\label{equ:Gamma} | \begin{equation}\label{equ:Gamma} | ||
- | {\bm \Gamma}_{IJ,n}=\beta \left \langle [\delta\mathbf{F}_{IJ}(n\Delta t)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle | + | {\Gamma}_{IJ,n}=\beta \left \langle [\delta\mathbf{F}_{IJ}(n\Delta t)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle |
\end{equation} | \end{equation} | ||
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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 ${\color{red}{Markovian\ assumption}}$ that the memory of fluctuating force in time is short enough to be approximated by a Dirac delta function |
\begin{eqnarray} | \begin{eqnarray} | ||
- | &\beta \langle [\delta\mathbf{F}_{IJ}(t-s)][\delta\mathbf{F}_{IJ}(0)]^T \rangle = 2 {\bm\gamma}_{IJ} \delta(t-s) \ , \label{equ:Mark_app1} \\ | + | &\beta \langle [\delta\mathbf{F}_{IJ}(t-s)][\delta\mathbf{F}_{IJ}(0)]^T \rangle = 2 {\gamma}_{IJ} \delta(t-s) \ , \label{equ:Mark_app1} \\ |
- | &{\beta}\int_{0}^{t} ds \left \langle [\delta\mathbf{F}_{IJ}(t-s)][\delta\mathbf{F}_{IJ}(0)]^T \right \rangle {\mathbf{V}_{IJ}(s)} = \bm\gamma_{IJ} \cdot {\mathbf{V}_{IJ}(t)} \ , \label{equ:Mark_app2} | + | &{\beta}\int_{0}^{t} ds \left \langle [\delta\mathbf{F}_{IJ}(t-s)][\delta\mathbf{F}_{IJ}(0)]^T \right \rangle {\mathbf{V}_{IJ}(s)} = \gamma_{IJ} \cdot {\mathbf{V}_{IJ}(t)} \ , \label{equ:Mark_app2} |
\end{eqnarray} | \end{eqnarray} | ||
- | where $\bm\gamma_{IJ}$ is the friction tensor defined by $\bm\gamma_{I J} = \beta \int_{0}^{\infty} dt \left \langle [\delta\mathbf{F}_{IJ}(t)][\delta\mathbf{F}_{IJ}(0)]^T \right \rangle $. Then, the equation of motion of DPD particles based on the Markovian approximation can be expressed by | + | where $\gamma_{IJ}$ is the friction tensor defined by $\gamma_{I J} = \beta \int_{0}^{\infty} dt \left \langle [\delta\mathbf{F}_{IJ}(t)][\delta\mathbf{F}_{IJ}(0)]^T \right \rangle $. Then, the equation of motion of DPD particles based on the Markovian approximation can be expressed by |
\begin{eqnarray}\label{equ:DPD} | \begin{eqnarray}\label{equ:DPD} | ||
\frac{d\mathbf{P}_I}{dt}&=&\sum_{J\neq I}\left\{\right. F^C_{IJ}(R_{IJ})\mathbf{e}_{IJ} | \frac{d\mathbf{P}_I}{dt}&=&\sum_{J\neq I}\left\{\right. F^C_{IJ}(R_{IJ})\mathbf{e}_{IJ} | ||
- | - {\bm\gamma}_{IJ}(R_{IJ}) \left( \mathbf{e}_{IJ}\cdot \mathbf{V}_{IJ} \right)\mathbf{e}_{IJ} | + | - {\gamma}_{IJ}(R_{IJ}) \left( \mathbf{e}_{IJ}\cdot \mathbf{V}_{IJ} \right)\mathbf{e}_{IJ} |
+\delta\mathbf{F}_{IJ}(t) \left.\right\} | +\delta\mathbf{F}_{IJ}(t) \left.\right\} | ||
\end{eqnarray} | \end{eqnarray} | ||