Tagged Particle Correlations in the Asymmetric Simple Exclusion Process: Finite Size Effects

TL;DR

This paper investigates finite size effects on the variance of a tagged particle's displacement in the ASEP, revealing oscillations due to sliding density fluctuations and analyzing their relation to universal fixed points in driven diffusive systems.

Contribution

It introduces a detailed analysis of finite size effects and oscillations in tagged particle variance in ASEP, connecting them to sliding density fluctuations and fixed point flows.

Findings

Variance grows linearly at small and large times

Oscillations occur at intermediate times with size-dependent period

Linearized model captures key qualitative features

Abstract

We study finite size effects in the variance of the displacement of a tagged particle in the stationary state of the Asymmetric Simple Exclusion Process (ASEP) on a ring of size $L$. The process involves hard core particles undergoing stochastic driven dynamics on a lattice. The variance of the displacement of the tagged particle, averaged with respect to an initial stationary ensemble and stochastic evolution, grows linearly with time at both small and very large times. We find that at intermediate times, it shows oscillations with a well defined size-dependent period. These oscillations arise from sliding density fluctuations (SDF) in the stationary state with respect to the drift of the tagged particle, the density fluctuations being transported through the system by kinematic waves. In the general context of driven diffusive systems, both the Edwards-Wilkinson (EW) and the Kardar-Parisi-Zhang (KPZ) fixed points are unstable with respect to the SDF fixed point, a flow towards which is generated on adding a gradient term to the EW and the KPZ time-evolution equation. We also study tagged particle correlations for a fixed initial configuration, drawn from the stationary ensemble, following earlier work by van Beijeren. We find that the time dependence of this correlation is determined by the dissipation of the density fluctuations. We show that an exactly solvable linearized model captures the essential qualitative features seen in the finite size effects of the tagged particle correlations in the ASEP. Moreover, this linearized model also provides an exact coarse-grained description of two other microscopic models.