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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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| \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 | ||
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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} | ||
