The equation of motion (EOM) of the coarse-grained (CG) particles obtained from the Mori-Zwanzig projection is given by

• Equation 1:

$$\begin{eqnarray}\label{equ:EoM} \frac{d}{dt}\mathbf{P}_I &=& \frac{1}{\beta} \frac{\partial}{\partial \mathbf{R}_I} {\rm{ln}} \omega(\mathbf{R}) \\ \nonumber &-& {\beta} \sum_{X=1}^{K}\int_{0}^{t} ds \left \langle [\delta\mathbf{F}_I(t-s)] [\delta\mathbf{F}_X(0)]^T \right \rangle \frac{\mathbf{P}_X(s)}{M_X} \\ &+& \delta\mathbf{F}_I(t) \ , \end{eqnarray}$$

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:

$$\begin{equation}\label{equ:omega} \omega (\mathbf{R})=\frac{\int d^N \mathbf{\hat r}\delta(\mathbf{\hat R}-\mathbf{R})e^{-\beta U} } {\int d^N \mathbf{\hat r} e^{-\beta U} } \ , \end{equation}$$

where $U$ is the potential energy corresponding to the phase point $\mathbf{R}$, and the integrations are performed over all the possible microscopic configurations $\{\mathbf{\hat r}_i\}$.

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 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 $\delta\mathbf{F}_I \approx \sum_{J\neq I}\delta\mathbf{F}_{IJ}$.

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 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.