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cg_zl [2014/08/21 21:41] zl25 |
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| \end{eqnarray} | \end{eqnarray} | ||
| \par | \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} | ||
| Line 84: | Line 84: | ||
| \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} | ||
