Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances

TL;DR

This paper studies the hydrodynamic limit of a one-dimensional subdiffusive exclusion process with random conductances, showing that the density evolves according to a generalized differential operator driven by an $ ext{a}$-stable subordinator.

Contribution

It introduces a quenched hydrodynamic limit for the process with random conductances in the domain of attraction of an $ ext{a}$-stable law, extending understanding of subdiffusive transport in disordered media.

Findings

Density profile converges to a solution of a generalized PDE involving a stable subordinator.

Established a law of large numbers for a tagged particle in the system.

Proved the hydrodynamic limit under quenched disorder conditions.

Abstract

Consider a system of particles performing nearest neighbor random walks on the lattice $\ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $\a$--stable law, $0<\a<1$. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function $L$, under the super-diffusive time scaling $N^{1 + 1/\alpha}L(N)$, the density profile evolves as the solution of the random equation $\partial_t \rho = \mf L_W \rho$, where $\mf L_W$ is the generalized second-order differential operator $\frac d{du} \frac d{dW}$ in which $W$ is a double sided $\a$--stable subordinator. This result follows from a quenched hydrodynamic limit in the case that the i.i.d. jump rates are replaced by a suitable array $\{\xi_{N,x} : x\in\bb Z\}$ having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.