Summary
We consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the initial measure isv ρ,λ , a product measure with densities ρ and λ to the left and right of the origin respectively, ρ<λ, then there exists a (microscopic) shock for the system. A shock is a random positionX t such that the system as seen from this position at timet has asymptotic product distributions with densities ρ and λ to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=lim t→∞ t −1(E(X t )2−(EX t )2) and findD=(p−q)(λ−ρ)−1(ρ(1−ρ)+λ(1−λ)) as conjectured by Spohn (1991). We show that in the scale\(\sqrt t\) the position ofX t is determined by the initial distribution of particles in a region of length proportional tot. 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 ρ and λ. 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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Ferrari, P.A., Fontes, L.R.G. Shock fluctuations in the asymmetric simple exclusion process. Probab. Th. Rel. Fields 99, 305–319 (1994). https://doi.org/10.1007/BF01199027
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DOI: https://doi.org/10.1007/BF01199027