diffusion tensor. Making use of the block-matrix inversion formula \[e.g. Equation (2.8.18) of Ref. \] we can check that the diffusion tensor fulfils the generalised Einstein relationships[@dickinson_brownian_1985; @peters_smoluchowski_2000] $$\begin{aligned}
  \beta\mathbf{D}_{\text{TT}} =& \left(\boldsymbol{\zeta}_{\text{TT}}-\boldsymbol{\zeta}_{\text{TR}}\boldsymbol{\zeta}_{\text{RR}}^{-1}\boldsymbol{\zeta}_{\text{RT}}\right)^{-1}\label{eq:40}\\
  \beta\mathbf{D}_{\text{TR}} =& -\left(\boldsymbol{\zeta}_{\text{TT}}-\boldsymbol{\zeta}_{\text{TR}}\boldsymbol{\zeta}_{\text{RR}}^{-1}\boldsymbol{\zeta}_{\text{RT}}\right)^{-1}\boldsymbol{\zeta}_{\text{TR}}\boldsymbol{\zeta}_{\text{RR}}^{-1}\equiv
  \left(\boldsymbol{\zeta}_{\text{RT}}-\boldsymbol{\zeta}_{\text{TT}}\boldsymbol{\zeta}_{\text{TR}}^{-1}\boldsymbol{\zeta}_{\text{RR}}\right)^{-1}\notag\\
  \beta\mathbf{D}_{\text{RT}} =& -\left(\boldsymbol{\zeta}_{\text{TT}}-\boldsymbol{\zeta}_{\text{TR}}\boldsymbol{\zeta}_{\text{RR}}^{-1}\boldsymbol{\zeta}_{\text{RT}}\right)^{-1}\boldsymbol{\zeta}_{\text{RT}}\boldsymbol{\zeta}_{\text{TT}}^{-1}
  \equiv\left(\boldsymbol{\zeta}_{\text{TR}}-\boldsymbol{\zeta}_{\text{RR}}\boldsymbol{\zeta}_{\text{RT}}^{-1}
  \boldsymbol{\zeta}_{\text{TT}}\right)^{-1}\notag\\
  \beta\mathbf{D}_{\text{RR}} =& \left(\boldsymbol{\zeta}_{\text{RR}}-\boldsymbol{\zeta}_{\text{RT}}\boldsymbol{\zeta}_{\text{TT}}^{-1}\boldsymbol{\zeta}_{\text{TR}}\right)^{-1}\notag
 \end{aligned}$$ In the special case when the translational and rotational motions are fully decoupled, $\mathbf{D}_{\text{TR}}=\mathbf{D}_{\text{RT}}\equiv 0$, we recover the equation derived by @rex_dynamical_2007. This is indeed the case for spherically isotropic colloids \[e.g. Equation (37) of Ref. \] far from a wall. Furthermore, if the colloidal particles are smooth spheres we recover the DDFT derived by @marconi_dynamic_2000, as the stress tensor is then uniquely determined by the translational component. However, the strength of the hydrodynamic translation-rotation coupling increases rapidly as the distance of the particles from a wall decreases[@dickinson_brownian_1985; @beenakker_many-sphere_1984]. The coupling is then unavoidable near the walls even for the simple spherical case. In this regard, Equation (\[eq:39\]) can be seen as a generalisation of the uncoupled DDFT obtained by @rex_dynamical_2007.

Of course, our overdamped DDFT (\[eq:39\]) does not include the self-propulsion forces present in the DDFT of @wittkowski_dynamical_2011 as these forces were not present in our original equations to begin with (see Appendix \[app:projection-operator-tech\]). 