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| In the right-hand side of Eq.(\ref{equ:EoM}), the first term represents the conservative force due to the change of microscopic configuration, and it is the ensemble average force on cluster $I$ denoted as $\langle\mathbf{F}_I \rangle$. The last term $\delta\mathbf{F}_I$ is the fluctuating force on cluster $I$ and it is given by $\delta\mathbf{F}_I = \mathbf{F}_I - \langle \mathbf{F}_I \rangle$ in which $\mathbf{F}_I$ is the instantaneous total force acting on the cluster $I$. The second term of Eq. (\ref{equ:EoM}) is the friction force determined by an integral of memory kernel. | In the right-hand side of Eq.(\ref{equ:EoM}), the first term represents the conservative force due to the change of microscopic configuration, and it is the ensemble average force on cluster $I$ denoted as $\langle\mathbf{F}_I \rangle$. The last term $\delta\mathbf{F}_I$ is the fluctuating force on cluster $I$ and it is given by $\delta\mathbf{F}_I = \mathbf{F}_I - \langle \mathbf{F}_I \rangle$ in which $\mathbf{F}_I$ is the instantaneous total force acting on the cluster $I$. The second term of Eq. (\ref{equ:EoM}) is the friction force determined by an integral of memory kernel. | ||
| - | {\color{red} First approximation:} Here, we assume that the non-bonded interactions between neighboring clusters in the microscopic system are explicitly {\bf pairwise decomposable}, and hence the total force consists of pairwise forces, e.g. $\mathbf{F}_I \approx \sum_{J\neq I}\mathbf{F}_{IJ}$ and $\delta\mathbf{F}_I \approx \sum_{J\neq I}\delta\mathbf{F}_{IJ}$. | + | ${\color{red}{First\ approximation:}}$ Here, we assume that the non-bonded interactions between neighboring clusters in the microscopic system are explicitly **pairwise decomposable**, and hence the total force consists of pairwise forces, e.g. $\mathbf{F}_I \approx \sum_{J\neq I}\mathbf{F}_{IJ}$ and $\delta\mathbf{F}_I \approx \sum_{J\neq I}\delta\mathbf{F}_{IJ}$. |
| - | \par | ||
| However, when we consider the force $\mathbf{F}_{I J}$ that a molecule $J$ exerts on another molecule $I$, in principle, $\mathbf{F}_{I J}$ involving many-body effects depends on all the COM coordinates $\mathbf{R}$ as well as their microscopic configurations. Although Eq. (\ref{equ:EoM}) based on the Mori-Zwanzig formalism is accurate, a direct computation of the many-body interactions is very difficult. | However, when we consider the force $\mathbf{F}_{I J}$ that a molecule $J$ exerts on another molecule $I$, in principle, $\mathbf{F}_{I J}$ involving many-body effects depends on all the COM coordinates $\mathbf{R}$ as well as their microscopic configurations. Although Eq. (\ref{equ:EoM}) based on the Mori-Zwanzig formalism is accurate, a direct computation of the many-body interactions is very difficult. | ||
| - | {\color{red} Second approximation:} In practice, we {\bf 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. |
