Summary
One considers a simple exclusion particle jump process on ℤ, where the underlying one particle motion is a degenerate random walk that moves only to the right. One starts with the configuration in which the left halfline is completely occupied and the right one free. It is shown that the number of particles at time t between site [u t] and [v t], divided by t, converges a.s. to \(\int\limits_u^\nu {f(w)dw}\), where f might be called the density profile. It is explicitely determined and shown to be an affine function. Secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u, converges weakly to the Bernoulli measure with density f(u), as the time tends to infinity.
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This work has been supported by the Deutsche Forschungsgemeinschaft
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Rost, H. Non-equilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrscheinlichkeitstheorie verw Gebiete 58, 41–53 (1981). https://doi.org/10.1007/BF00536194
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DOI: https://doi.org/10.1007/BF00536194