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| ${\color{red}{Second\ approximation:}}$ In practice, we neglect the many-body correlations between different pairs, and assume that the force $\mathbf{F}_{I J}$ between two clusters $I$ and $J$ depends only on the relative COM positions $\mathbf{R}_{I}$ and $\mathbf{R}_{J}$ and is independent of the positions of the rest of clusters. | ${\color{red}{Second\ approximation:}}$ In practice, we neglect the many-body correlations between different pairs, and assume that the force $\mathbf{F}_{I J}$ between two clusters $I$ and $J$ depends only on the relative COM positions $\mathbf{R}_{I}$ and $\mathbf{R}_{J}$ and is independent of the positions of the rest of clusters. | ||
| + | |||
| + | Therefore, the conservative term is in Eq.(\ref{equ:EoM}) approximated by | ||
| + | \begin{equation}\label{equ:FC} | ||
| + | \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} | ||
| + | \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. | ||
| + | |||
| + | \par | ||
| + | Similarly, the fluctuating force defined as the deviation from the mean force is also decomposed into pairwise forces | ||
| + | \begin{eqnarray}\label{equ:FR} | ||
| + | \delta\mathbf{F}_I &=& \mathbf{F}_I - \langle \mathbf{F}_I \rangle \approx \sum_{J\neq I}\delta\mathbf{F}_{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} | ||
| + | \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}). | ||
| + | |||
| + | \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 | ||
| + | \begin{eqnarray}\label{equ:no_corelation} | ||
| + | &&\sum_{J\neq I}\sum_{Y\neq X}\left \langle [\delta\mathbf{F}_{IJ}(t-s)] [\delta\mathbf{F}_{XY}(0)]^T \right \rangle \mathbf{V}_X(s) \nonumber\\ | ||
| + | &=&\left \langle [\delta\mathbf{F}_{IJ}(t-s)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle \mathbf{V}_I(s)|_{X=I,Y=J} + \nonumber\\ | ||
| + | &&\left \langle [\delta\mathbf{F}_{IJ}(t-s)] [\delta\mathbf{F}_{JI}(0)]^T \right \rangle \mathbf{V}_J(s)|_{X=J,Y=I} \nonumber\\ | ||
| + | &=&\left \langle [\delta\mathbf{F}_{IJ}(t-s)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle \mathbf{V}_{IJ}(s) | ||
| + | \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 $N\Delta t$ | ||
| + | \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 | ||
| + | \end{equation} | ||
| + | |||
| + | Then, the second term in Eq.(\ref{equ:EoM}) can be expanded as follows: | ||
| + | |||
| + | \begin{eqnarray}\label{equ:FD} | ||
| + | &&-{\beta} \sum_{X=1}^{K}\int_{0}^{t} ds \left\langle [\delta\mathbf{F}_I(t-s)] [\delta\mathbf{F}_X(0)]^T \right \rangle \mathbf{V}_X(s) \nonumber\\ | ||
| + | &=& -{\beta}\sum_{X=1}^{K}\int_{t-N\Delta t}^{t} ds \left \langle [\delta\mathbf{F}_I(t-s)] [\delta\mathbf{F}_X(0)]^T \right \rangle \mathbf{V}_X(s) \nonumber\\ | ||
| + | &=& -{\beta\cdot \Delta t}\sum_{X=1}^{K}\sum_{n=0}^{N}\alpha_n \left \langle [\delta\mathbf{F}_{I}(n\Delta t)] [\delta\mathbf{F}_{X}(0)]^T \right \rangle \mathbf{V}_X(t-n\Delta t) \nonumber\\ | ||
| + | &=& -{\beta\cdot \Delta t}\sum_{X=1}^{K}\sum_{J\neq I}\sum_{Y\neq X}\sum_{n=0}^{N}\alpha_n \left \langle [\delta\mathbf{F}_{IJ}(n\Delta t)] [\delta\mathbf{F}_{XY}(0)]^T \right \rangle \mathbf{V}_X(t-n\Delta t) \nonumber\\ | ||
| + | &=& -{\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} | ||
| + | \par | ||
| + | We define the friction matrix ${\Gamma}_{IJ,n}$ as | ||
| + | \begin{equation}\label{equ:Gamma} | ||
| + | {\Gamma}_{IJ,n}=\beta \left \langle [\delta\mathbf{F}_{IJ}(n\Delta t)] [\delta\mathbf{F}_{IJ}(0)]^T \right \rangle | ||
| + | \end{equation} | ||
| + | |||
| + | where $\delta\mathbf{F}_{IJ}$ is the fluctuating force. Generally, the fluctuating force $\delta {\bf{F}}_{I J}$ is not parallel to the radial direction $\mathbf{e}_{IJ}$. However, $\delta {\bf{F}}_{I J}$, on average, is transversely isotropic with respect to $\mathbf{e}_{IJ}$ because the instantaneous pairwise force $\mathbf{F}_{IJ}$ has no preference between directions $\perp_1$ and $\perp_2$. | ||
| + | |||
| + | When we calculate the friction matrix, we do not distinguish between the directions $\perp_1$ and $\perp_2$ and decompose $\delta\mathbf{F}_{IJ}$ into two parts | ||
| + | |||
| + | \begin{align} | ||
| + | \delta \mathbf{F}_{IJ}&=(\mathbf{e}_{IJ}\mathbf{e}^T_{IJ})\cdot\delta\mathbf{F}_{IJ}+(\mathbf{1}- | ||
| + | \mathbf{e}_{IJ}\mathbf{e}^T_{IJ})\cdot\delta\mathbf{F}_{IJ} \nonumber \\ &=\delta\mathbf{F}^{\parallel}_{IJ}+\delta\mathbf{F}^{\perp}_{IJ} \ , \label{equ:randomF} | ||
| + | \end{align} | ||
| + | |||
| + | 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 | ||
| + | |||
| + | \begin{eqnarray} | ||
| + | &\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)} = \gamma_{IJ} \cdot {\mathbf{V}_{IJ}(t)} \ , \label{equ:Mark_app2} | ||
| + | \end{eqnarray} | ||
| + | |||
| + | 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} | ||
| + | \frac{d\mathbf{P}_I}{dt}&=&\sum_{J\neq I}\left\{\right. F^C_{IJ}(R_{IJ})\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\} | ||
| + | \end{eqnarray} | ||
| + | |||
