Large deviation function of a tracer position in single file diffusion

TL;DR

This paper derives the large deviation function for a tracer's displacement in single file diffusion, revealing how it can be computed via a non-interacting particle mapping and extending understanding of multi-time correlations.

Contribution

It introduces a method to compute the large deviation function for tracer displacement in single file diffusion using a mapping to non-interacting particles, and explores the relation between quenched and annealed distributions.

Findings

Large deviation function can be obtained through a non-interacting particle mapping.

Confirmed previous results on one-time displacement distribution.

Established a relation between tracer and current distributions in quenched case.

Abstract

Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.