Macroscopic fluctuations of a driven tracer in the symmetric exclusion process

TL;DR

This paper investigates the macroscopic fluctuations of a driven tracer in the symmetric exclusion process, using hydrodynamic limits and fluctuation theory to derive key statistical properties and compare with microscopic results.

Contribution

It introduces a mapping to the zero-range process to analyze the tracer dynamics and derives the cumulant generating function using Macroscopic Fluctuation Theory.

Findings

Derived the average displacement of the tracer as a function of bias and densities.

Obtained the cumulant generating function in the high-density limit.

Showed tracer variance depends on initial condition susceptibility.

Abstract

The dynamics of an asymmetric tracer in the symmetric simple exclusion process (SEP) is mapped, in the continuous scaling limit, to the local current through the origin in the zero-range process (ZRP) with a biased bond. This allows us to study the hydrodynamics of the SEP with an asymmetric tracer with a step initial condition, leading to the average displacement as a function of the bias and the densities on both sides. We then derive the cumulant generating function of the process in the high-density limit, by using the Macroscopic Fluctuation Theory and obtain agreement with the microscopic results of Poncet et al (2021). For more general initial conditions, we show that the tracer variance in the high-density limit depends only on the generalized susceptibility in the initial condition.