Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models

Abstract

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^βρ_t}{\partial t^β} = Δρ_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 ρ_t}{\partial t}= \mathcal L_βρ_t\ , \end{equation*} where ${\mathcal{L}}_{\beta }$ is the random generator of the singular quasi-diffusion known as FIN diffusion.