Shock fluctuations in asymmetric simple exclusion. (English) Zbl 0744.60117

Ferrari, Pablo A.

Probab. Theory Relat. Fields 91, No. 1, 81-101 (1992).

The one-dimensional nearest neighbors asymmetric simple exclusion process is used as a microscopic approximation to the Burgers equation. We study the process with rates of jumps \(p>q\) to the right and left, respectively, and with initial product measure with densities \(\rho<\lambda\) to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density \((\rho=0)\) we prove, in the scale \(t^{1/2}\), that the position of the shock at time \(t\) depends only on the initial configuration in a region depending on \(t\). The proofs are based on laws of large numbers for the second class particle.

Reviewer: P.A.Ferrari (São Paulo)

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

asymmetric simple exclusion process; Burgers equation; laws of large numbers

References:

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