# Differences

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 cg_zl [2014/08/21 17:42]zl25 cg_zl [2014/08/21 17:43]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 Last revision Both sides next revision Line 29: Line 29: \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} Line 38: Line 37: ​\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} Line 49: Line 47: &​=&​\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} Line 64: Line 62: &=& -{\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}  