Fluctuations and correlations of a driven tracer in a hard-core lattice gas

TL;DR

This paper develops an analytical framework to study the fluctuations and correlations of a driven tracer particle in a hard-core lattice gas, revealing non-monotonic behavior of its dispersion with respect to bath particle density.

Contribution

It introduces an exact analytical approach that extends beyond standard force-velocity relations, providing detailed insights into tracer fluctuations and correlations in active microrheology.

Findings

Exact Einstein-type relation linking fluctuations and density profiles

Explicit expression for tracer position variance in 1D

Discovery of non-monotonic dispersion behavior with density

Abstract

We consider a driven tracer particle (TP) in a bath of hard-core particles undergoing continuous exchanges with a reservoir. We develop an analytical framework which allows us to go beyond the standard force-velocity relation used for this minimal model of active microrheology and quantitatively analyze, for any density of the bath particles, the fluctuations of the TP position and their correlations with the occupation number of the bath particles. We obtain an exact Einstein-type relation which links these fluctuations in the absence of a driving force and the bath particles density profiles in the linear driving regime. For the one-dimensional case we also provide an approximate but very accurate explicit expression for the variance of the TP position and show that it can be a non-monotoneous function of the bath particles density: counter-intuitively, an increase of the density may increase the dispersion of the TP position. We show that this non trivial behavior, which could in principle be observed in active microrheology experiments, is induced by subtle cross-correlations quantified by our approach.