Bulk diffusion of 1D exclusion process with bond disorder

TL;DR

This paper studies how a one-dimensional exclusion process with bond disorder diffuses over time, showing that the macroscopic behavior follows the heat equation with a diffusion constant related to the average jump rate.

Contribution

It extends existing methods to analyze the hydrodynamic limit of exclusion processes with bond disorder, establishing diffusive and subdiffusive regimes based on the average jump rate.

Findings

Diffusive behavior when the average jump rate A is finite.

Subdiffusive behavior when A is infinite.

Hydrodynamic limit described by the heat equation with diffusion constant 1/A.

Abstract

Given a doubly infinite sequence of positive numbers {c_k: k in Z} satisfying a LLN with limit A, we consider the nearest-neighbor simple exclusion process on Z where c_k is the probability rate of jumps between k and k+1. If A is infinite we require an additional minor technical condition. By extending a method developed by K. Nagy, we show that the diffusively rescaled process has hydrodynamic behavior described by the heat equation with diffusion constant 1/A. In particular, the process has diffusive behavior for finite A and subdiffusive behavior for infinite A.