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 cg_zl [2014/08/21 17:43]zl25 cg_zl [2014/08/21 17:45] (current)zl25 Both sides previous revision Previous revision 2014/08/21 17:45 zl25 2014/08/21 17:43 zl25 2014/08/21 17:42 zl25 2014/08/21 17:41 zl25 2014/08/21 17:36 zl25 2014/08/21 17:28 zl25 created 2014/08/21 17:45 zl25 2014/08/21 17:43 zl25 2014/08/21 17:42 zl25 2014/08/21 17:41 zl25 2014/08/21 17:36 zl25 2014/08/21 17:28 zl25 created Line 1: Line 1: ===== 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 Line 10: Line 10: 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} } \ , Line 48: Line 47: \end{eqnarray} \end{eqnarray} - 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 Line 79: Line 78: 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}  