 9256 Janowsky Steven A., Lebowitz Joel L.
 Finite Size Effects and Shock Fluctuations in the Asymmetric Simple
Exclusion Proces
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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 $1p$), $1/2<p\leq 1$. 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 $r \lessapprox (12\rho_{\rm avg}1)/ (1+2\rho_{\rm
avg}1)$ 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 welldefined
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 particlehole
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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