Equilibrium Perturbations for Asymmetric Zero Range Process under Diffusive Scaling in Dimensions $d \geq 2$

TL;DR

This paper studies the asymmetric zero range process in higher dimensions, showing that small initial perturbations evolve according to the heat equation under diffusive scaling, given certain jump rate conditions.

Contribution

It demonstrates that under specific constraints, the macroscopic behavior of perturbed asymmetric zero range processes follows the heat equation in dimensions $d \\geq 2$.

Findings

Perturbed density profiles evolve as solutions to the heat equation.

The results hold under particular conditions on jump rates.

The analysis applies to initial perturbations of order $N^{-\\alpha}$, with $\\alpha \\in (0,1)$.

Abstract

We consider the asymmetric zero range process in dimensions $d \geq 2$. Assume the initial density profile is a perturbation of the constant density, which has order $N^{-\alpha}$, $\alpha \in (0,1)$, and is constant along the drift direction. Here, $N$ is the scaling parameter. We show that under some constraints on the jump rate of the zero range process, the perturbed quantity macroscopically obeys the heat equation under diffusive scaling.