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

TL;DR

This paper derives a fractional kinetics equation as a hydrodynamic limit for a Markovian particle system in a random trapping environment, revealing sub-diffusive behavior in higher dimensions and a different limit in one dimension.

Contribution

It establishes a connection between a Markovian interacting particle system and fractional kinetics equations, highlighting dimension-dependent limiting behaviors.

Findings

In dimensions d≥2, the empirical density converges to a fractional diffusion equation.

In dimension d=1, the system converges to a solution involving the FIN diffusion generator.

The study demonstrates the emergence of non-Markovian dynamics from Markovian particle systems.

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^\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.