  We consider a partial exclusion process evolving on $\mathbb Z^d$ in a random
trapping environment. In dimension $d\ge 2$, we derive the fractional kinetics
equation \begin{equation*}\frac{\partial^\beta\rho_t}{\partial t^\beta} =
\Delta \rho_t
  \end{equation*} as a hydrodynamic limit of the particle system. Here,
$\frac{\partial^\beta}{\partial t^\beta}$, $\beta\in(0,1)$, denotes the
fractional derivative in the Caputo sense. We thus exhibit a Markovian
interacting particle system whose empirical density field rescales to a
sub-diffusive equation corresponding to a non-Markovian process, the Fractional
Kinetics process. In contrast, we show that, when $d=1$, the system rescales to
the solution to \begin{equation*}
  \frac{\partial \rho_t}{\partial t}= \mathcal L_\beta \rho_t\ ,
\end{equation*} where $\mathcal L_\beta$ is the random generator of the
singular quasi-diffusion known as FIN diffusion.
