Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
cg_zl [2014/08/21 17:43]
zl25
cg_zl [2014/08/21 17:45] (current)
zl25
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, ​${\color{red}{Thirdapproximation:​}}$ 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
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 ${\color{red}{Markovianassumption}}$ that the memory of fluctuating force in time is short enough to be approximated by a Dirac delta function
  
 \begin{eqnarray} \begin{eqnarray}

Navigation
QR Code
QR Code Formulation of MZ-guided Markovian/Non-Markovian DPD (generated for current page)