Symmetric Exclusion Process with a Localized Source

TL;DR

This paper studies how particles accumulate over time in a symmetric exclusion process with a localized source, revealing dimension-dependent growth laws and fluctuation behaviors.

Contribution

It provides a detailed analysis of the asymptotic growth of particles and fluctuations in different dimensions for the first time.

Findings

Growth of particles as t^{1/2} in 1D

Growth as t/log(t) in 2D

Linear growth in higher dimensions

Abstract

We investigate the growth of the total number of particles in a symmetric exclusion process driven by a localized source. The average total number of particles entering an initially empty system grows with time as t^{1/2} in one dimension, t/log(t) in two dimensions, and linearly in higher dimensions. In one and two dimensions, the leading asymptotic behaviors for the average total number of particles are independent on the intensity of the source. We also discuss fluctuations of the total number of particles and determine the asymptotic growth of the variance in one dimension.