Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process.

Abstract

We consider a system of particles on a lattice of L sites, set on a circle, evolving according to the asymmetric simple-exclusion process, i.e., particles jump independently to empty neighboring sites on the right (left) with rate p (rate 1-p), 1/2p\ensuremath{\le}1. We study the nonequilibrium stationary states of the system, when the translation invariance is broken by the insertion of a blockage between (say) sites L and 1; this reduces the rates at which particles jump across the bond by a factor r, 0r1. For fixed overall density ${\mathrm{\ensuremath{\rho}}}_{\mathrm{avg}}$ and r\ensuremath{\lesssim}(1-\ensuremath{\Vert}2${\mathrm{\ensuremath{\rho}}}_{\mathrm{avg}}$-1\ensuremath{\Vert})/(1+\ensuremath{\Vert}2${\mathrm{\ensuremath{\rho}}}_{\mathrm{avg}}$-1\ensuremath{\Vert}), this causes the system to segregate into two regions with densities ${\mathrm{\ensuremath{\rho}}}_{1}$ and ${\mathrm{\ensuremath{\rho}}}_{2}$=1-${\mathrm{\ensuremath{\rho}}}_{1}$, where the densities depend only on r and p, with the two regions separated by a well-defined sharp interface. This corresponds to the shock front described macroscopically in a uniform system by the Burgers equation. We find that fluctuations of the shock position about its average value grow like ${\mathit{L}}^{1/2}$ or ${\mathit{L}}^{1/3}$, depending upon whether particle-hole symmetry exists. This corresponds to the growth in time of ${\mathit{t}}^{1/2}$ and ${\mathit{t}}^{1/3}$ of the displacement of a shock front from the position predicted by the solution of the Burgers equation in a system without a blockage and provides an alternative method for studying such fluctuations.