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

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), 1/2<p1. 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, 0<r<1. For fixed overall density avg and r(1-∥2avg-1∥)/(1+∥2avg-1∥), this causes the system to segregate into two regions with densities 1 and 2=1-1, 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 L1/2 or L1/3, depending upon whether particle-hole symmetry exists. This corresponds to the growth in time of t1/2 and t1/3 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.