Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond

TL;DR

This paper analyzes the hydrodynamic behavior of a totally asymmetric exclusion process with a slow bond, deriving a variational formula for the macroscopic profile valid across all slow rates.

Contribution

It introduces a hydrodynamic limit for the process with a slow bond, using a variational coupling method, applicable to various initial conditions including deterministic ones.

Findings

Hydrodynamic limit described by a variational formula

Macroscopic rate at the origin exceeds microscopic slow rate

Valid for all slow rate values

Abstract

We study a totally asymmetric simple exclusion process where jumps happen at rate one, except at the origin where the rate is lower. We prove a hydrodynamic scaling limit to a macroscopic profile described by a variational formula. The limit is valid for all values of the slow rate. The only assumption required is that a law of large numbers holds for the initial particle distribution. This includes also deterministic initial configurations. The hydrodynamic description contains as an unknown parameter the macroscopic rate at the origin, which is strictly larger than the microscopic slow rate. The limit is proved by the variational coupling method.