Nevertheless, they can readily be included as an additional term, ${\boldsymbol{\mathfrak{F}}}_{i}^{A}({\boldsymbol{\mathfrak{r}}}_i;t)$, inside the solvent-averaged interaction forces and torques ${\boldsymbol{\mathfrak{F}}}$. This results in Equation (\[eq:11\]) having the extra term, $-{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{F}}}_{1}^{A}({\boldsymbol{\mathfrak{r}}}_1;t)\rho({\boldsymbol{\mathfrak{r}}}_1;t)$. Following exactly the same steps as in (\[eq:12\])–(\[eq:16\]), the same first equation of the YBG hierarchy (\[eq:17\]) can be obtained, by setting ${\boldsymbol{\mathfrak{F}}}_1^A=0$ at equilibrium[@wittkowski_dynamical_2011]. Therefore, the additional term $-{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{F}}}_{1}^{A}({\boldsymbol{\mathfrak{r}}}_1;t)\rho({\boldsymbol{\mathfrak{r}}}_1;t)$ would survive throughout the whole derivation so that the left-hand side of our DDFT equation (\[eq:34\]) would contain it. Eventually, this leads us to $$\begin{aligned}
\partial_t\rho({\boldsymbol{\mathfrak{r}}}_1;t)={\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot\left(\rho({\boldsymbol{\mathfrak{r}}}_1;t)\beta\mathbf{D}({\boldsymbol{\mathfrak{r}}}_1;t)\left({\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}-{\boldsymbol{\mathfrak{F}}}_{1}^{A}({\boldsymbol{\mathfrak{r}}}_1;t)\right)\right)\label{eq:41}\end{aligned}$$ which is now in complete agreement with the equation previously derived within the overdamped regime[@wittkowski_dynamical_2011], with the diffusion tensor still satisfying the generalised Einstein relationships (\[eq:40\]).

Conclusions {#sec:conclusions}
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In this work we have formulated a DDFT for orientable particles with inertia and HIs. There are numerous examples where orientation of particles plays a key role. From the fundamental study of perfectly-rough spheres[@condiff_brownian_1966], loaded spherocylinders[@condiff_brownian_1966; @dahler_kinetic_1963], nematic solutions[@hartel_towing_2010; @zhang_isotropic-nematic_2006] or liquid-crystal nucleation[@schilling_self-poisoning_2004; @zhang_isotropic-nematic_2006; @miller_hierarchical_2009], to the study of biological processes such as[@dickinson_brownian_1985] protein adsorption and trapping, antibody-antigen interaction, biochemical assembly by monomer aggregation or polymerization, bone formation[@cantaert_nanoscale_2013; @gomez-morales_progress_2013] or *in vivo* protein crystallization[@bechtel_electron_1976; @koopmann_vivo_2012], to name but a few. We believe that all 