Superdiffusive Scaling Limits for the Symmetric Exclusion Process with Slow Bonds

TL;DR

This paper explores superdiffusive scaling limits of the symmetric exclusion process with slow bonds, revealing new regimes where the system exhibits either static or evolving density profiles, culminating in a continuous heat equation as the number of slow bonds grows.

Contribution

It introduces novel superdiffusive scaling limits for the symmetric exclusion process with multiple slow bonds, extending understanding beyond the previously studied diffusive regime.

Findings

For fixed k and time scale k^2 n^θ with θ in (2, 1+β), the density remains constant in each box.

At time scale k^2 n^{1+β}, the density evolves according to a discrete heat equation.

When k increases to infinity at the same time scale, the system converges to the continuous heat equation on the torus.

Abstract

In \cite{fgn1}, the hydrodynamic limit in the diffusive scaling of the symmetric simple exclusion process with a finite number of slow bonds of strength $n^{-\beta}$ has been studied. Here $n$ is the scaling parameter and $\beta>0$ is fixed. As shown in \cite{fgn1}, when $\beta>1$, such a limit is given by the heat equation with Neumann boundary conditions. In this work, we find more non-trivial super-diffusive scaling limits for this dynamics. Assume that there are $k$ equally spaced slow bonds in the system. If $k$ is fixed and the time scale is $k^2n^\theta$, with $\theta\in (2,1+\beta)$, the density is asymptotically constant in each of the $k$ boxes, and equal to the initial expected mass in that box, i.e., there is no time evolution. If $k$ is fixed and the time scale is $k^2n^{1+\beta}$, then the density is also spatially constant in each box, but evolves in time according to the discrete heat equation. Finally, if the time scale is $k^2n^{1+\beta}$ and, additionally, the number of boxes $k$ increases to infinity, then the system converges to the continuous heat equation on the torus, with no boundary conditions.