bath, eventually invalidating the inertial LEs. Nevertheless, while the inertial equations (\[eq:1\]) are neither formally exact nor completely general, it can be shown that there exist very sensible regimes in which neglecting inertia in the bath is appropriate, namely when the colloidal-particle density, $\rho_B$, is much larger than that of the bath particles, $\rho_b$. This is done in Appendix \[app:neglecting-bath-inertia\] using dimensional analysis showing that there is indeed a mass-induced time-scale separation when $\rho_b/\rho_B\ll1$. Under such circumstances the LEs and the associated FPE must be applicable, as explicitly pointed out also by @roux_brownian_1992.

From the system of stochastic differential equations (SDEs) in Eq. (\[eq:2\]), one can obtain[@risken_fokker-planck_1996] (see Appendix \[app:projection-operator-tech\]) the time-evolution equation of the phase-space probability density function (PDF), ${f^{(N)}}({\boldsymbol{\mathfrak{r}}}^N,{\boldsymbol{\mathfrak{p}}}^N;t)$, to find each particle $j$ with positions ${\mathbf{r}}_j,\ {\boldsymbol{\alpha}}_j$ and momenta ${\mathbf{p}}_j,\ {\mathbf{L}}_j$ at time $t$, $$\begin{aligned}
\partial_t {f^{(N)}}({\boldsymbol{\mathfrak{r}}}^N,{\boldsymbol{\mathfrak{p}}}^N;t) +&
\sum_{j=1}^N{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_j}\cdot \big({\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_j{f^{(N)}}({\boldsymbol{\mathfrak{r}}}^N,{\boldsymbol{\mathfrak{p}}}^N;t) \big)+\sum_{j=1}^N{\boldsymbol{\mathfrak{F}}}_j\cdot{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{p}}}_j}{f^{(N)}}({\boldsymbol{\mathfrak{r}}}^N,{\boldsymbol{\mathfrak{p}}}^N;t) \label{eq:4}\\
&= \sum_{j,k=1}^N \,{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{p}}}_j}\cdot{\boldsymbol{\Gamma}}_{jk}({\boldsymbol{\mathfrak{r}}}^N)\left({\boldsymbol{\mathfrak{p}}}_k+k_BT{\boldsymbol{\mathfrak{m}}}\,{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{p}}}_k}\right){f^{(N)}}({\boldsymbol{\mathfrak{r}}}^N,{\boldsymbol{\mathfrak{p}}}^N;t)\notag.\end{aligned}$$ It is well-known that Equations (\[eq:2\]) or (\[eq:4\]) are computationally tractable only for small systems. This problem can be circumvented to an extent by assuming a very simple structure for the friction tensor, namely ${\boldsymbol{\Gamma}}= \text{diag}(\gamma_{TT}\mathbf{1},\gamma_{RR}\mathbf{1})$. While such an approximation certainly reduces the numerical complexity \[since determining $\mathbf{A}$ is now $O(N)$ rather 