than $O(N^3)$\], it physically corresponds to ignoring HIs, thus neglecting crucial physical information. Another approach to tackle the problem, which is the one we adopt, is to derive a coarse-grained/mean-field model by averaging over the degrees of freedom of all but a few particles[@wu_density-functional_2007; @chan_time-dependent_2005; @goddard_unification_2013; @goddard_multi-species_2013]. This method yields a lower-dimensional system but requires knowledge of the functional relation between ${f^{(N)}}$ and the reduced distribution function. Therefore, convenient and accurate closures for the dynamics of the reduced distribution function must be found.

Continuity and flux equations \[sec:continuity-current-equation\]
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Given (\[eq:4\]), we can reduce to a lower dimensional problem by averaging over all but a few particles. Our first assumption involves the solvent-averaged interaction potential $V({\boldsymbol{\mathfrak{r}}}^N)$. Inspired by the decomposition proposed for spherical particles[@goddard_unification_2013] we assume that this potential can be split into linear combinations of $n$-particle interactions, i.e. $$\begin{aligned}
 V({\boldsymbol{\mathfrak{r}}}^N;t)= \sum_{j=1}^NV_1({\boldsymbol{\mathfrak{r}}}_j;t)+\sum_{n=2}^N\frac{1}{n!}\sum_{k_1\neq\dots\neq k_n=1}^n V_n({\boldsymbol{\mathfrak{r}}}_{{k}_1},\dots,{\boldsymbol{\mathfrak{r}}}_{{k}_n})\label{eq:5}.\end{aligned}$$ For ease of notation we also adopt the decomposition ${\boldsymbol{\Gamma}}_{ij}^{\mu\nu}=\gamma_{\mu\nu}\delta_{ij}\mathbf{1}+\gamma_{\mu\nu}\widetilde{{\boldsymbol{\Gamma}}}_{ij}^{\mu\nu}$, recalling that we are not making use of Einstein’s summation convention. Thus, $$\begin{aligned}
 {\boldsymbol{\Gamma}}_{ij}=\delta_{ij}{\boldsymbol{\Upsilon}}+\widetilde{{\boldsymbol{\Upsilon}}}_{ij} ,\end{aligned}$$ with the $6\times 6$ tensor ${\boldsymbol{\Upsilon}}= (\gamma_{\mu\nu}\mathbf{1})$, where $\gamma_{\mu\nu}$ is the constant friction coefficient for an isolated particle and $\widetilde{{\boldsymbol{\Upsilon}}}_{ij}=(\gamma_{\mu\nu}\widetilde{{\boldsymbol{\Gamma}}}_{ij}^{\mu\nu})$, where $\widetilde{{\boldsymbol{\Gamma}}}_{ij}^{\mu\nu}$ are the translational-rotational HI tensors. In the special case 