93-29 P.A.Ferrari , L.R.G.Fontes

Shock fluctuations in the asymmetric simple exclusion process (46K, TeX) Feb 11, 93

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Abstract. We consider the one dimensional nearest neighbors asymmetric simple exclusion process with rates $q$ and $p$ for left and right jumps respectively; $q<p$. Ferrari, Kipnis and Saada (1991) have shown that if the initial measure is $\nu_{\rho,\lambda}$, a product measure with densities $\rho$ and $\lambda$ to the left and right of the origin respectively, $\rho<\lambda$, then there exists a (microscopic) shock for the system. A shock is a random position $X_t$ such that the system as seen from this position at time $t$ has asymptotic product distributions with densities $\rho$ and $\lambda$ to the left and right of the origin respectively, uniformly in $t$. We compute the diffusion coefficient of the shock $D=\lim_{t\to\infty} t^{-1}(E(X_t)^2 - (EX_t)^2)$ and find $D=(p-q)(\lambda-\rho)^{-1} (\rho(1-\rho)+\lambda(1-\lambda))$ as conjectured by Spohn (1991). We show that in the scale $\sqrt t$ the position of $X_t$ is determined by the initial distribution of particles in a region of lenght proportional to $t$. We prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities $\rho$ and $\lambda$. This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.

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