92-56 Janowsky Steven A., Lebowitz Joel L.
Finite Size Effects and Shock Fluctuations in the Asymmetric Simple Exclusion Proces (177K, LaTeX) May 15, 92
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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, {\em i.e.}\ particles jump independently to empty neighboring sites on the right (left) with rate $p$ (rate $1-p$), $1/2<p\leq 1$. 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 $r$, $0<r<1$. For fixed overall density $\rho_{\rm avg}$ and $r \lessapprox (1-|2\rho_{\rm avg}-1|)/ (1+|2\rho_{\rm avg}-1|)$ this causes the system to segregate into two regions with densities $\rho_1$ and $\rho_2=1-\rho_1$, 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 $L^{1/2}$ or $L^{1/3}$, depending upon whether particle-hole symmetry exists. This corresponds to the growth in time of $t^{1/2}$ and $t^{1/3}$ 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 a new method for studying such fluctuations.

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