an existence and uniqueness theorem for the Smoluchowski equation with non-constant diffusion tensor and effective drift vector dependent on the two-body HI tensors $\bm{Z}_1$ and $\bm{Z}_2$. In Section \[sec:discussion\] we present our concluding remarks and state some open problems. In Appendix \[sec:classical\_paraboliv\_pde\] we provide some technical results that are used in the proof of Theorem \[thm:exis\_uniq\_weak\_sol\_rho\]. Finally in Appendix \[app:nomenclature\] we provide a list of nomenclature.

Preliminaries {#sec:preliminaries}
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In this section we specify the nonlinear boundary conditions and initial data for the DDFT . We also nondimensionalise the governing equations and provide the assumptions on the regularity of the potentials, correlation function, diffusion tensor and initial data.

Boundary Conditions. {#subsec:boundary_conditions}
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When $U = \mathbb{R}^d$ we take $$\begin{aligned}
\label{bc:density_and_flux_decaying}
\left\{\begin{aligned}
\varrho(\vec{r},t)\to 0 \\
\vec{a}(\vec{r},t)\to \vec{0}
\end{aligned}\right. \quad \text{ as } \quad |\vec{r}|\to \infty,\end{aligned}$$ where we require $V_1$ to be growing at least quadratically as $\vec{r}\to \infty$. Physically-speaking this prevents the density from running out to infinity. When $U\subset \mathbb{R}^d$ is open and bounded we impose that the total mass of the system $M$ remains constant, in particular we have $$\begin{aligned}
\label{bc:mass_preserving}
\vec{a}(\vec{r},t)\cdot\vec{n}\bigg|_{\partial U\times [0,T]} = 0.\end{aligned}$$ The boundary condition may be viewed as a [*nonlinear*]{} Robin condition imposing the 