Distribution of the position of a driven tracer in a hardcore lattice gas

TL;DR

This paper analyzes the distribution and fluctuations of a biased tracer particle in a hardcore lattice gas, revealing non-monotonic diffusion behavior in one dimension and deriving the full probability distribution of the tracer's position.

Contribution

It extends previous mean-field approximations to higher dimensions and provides a detailed solution for the one-dimensional case, including the distribution of the tracer's position.

Findings

Diffusion coefficient is non-monotonic with respect to bath particle density in 1D.

The position distribution of the tracer becomes Gaussian at long times.

The extended approximation applies to any space dimension.

Abstract

We study the position of a biased tracer particle (TP) in a bath of hardcore particles moving on a lattice of arbitrary dimension and in contact with a reservoir. Starting from the master equation satisfied by the joint probability of the TP position and the bath configuration and resorting to a mean-field-type approximation, we presented a computation of the fluctuations of the TP position in a previous publication [Phys. Rev. E \textbf{87}, 032164 (2013)]. Counterintuitively, on a one-dimensional lattice, the diffusion coefficient of the TP was shown to be a non-monotonic function of the density of bath particles, and reaches a maximum for a nonzero value of the density. Here, we: (i) give the details of this computation and offer a physical insight into the understanding of the non-monotonicity of the diffusion coefficient; (ii) extend the mean-field-type approximation to decouple higher-order correlation functions, and obtain the evolution equation satisfied by the cumulant generating function of the position of the TP, valid in any space dimension. In the particular case of a one-dimensional lattice, we solve this equation and obtain the probability distribution of the TP position. We show that the position rescaled by its fluctuations is asymptotically distributed accordingly to a Gaussian distribution in the long-time limit.