Shocks in asymmetric one-dimensional exclusion processes

Abstract

We review recent results concerning the local
structure of the shocks in the one dimensional nearest neighbors totally asymmetric simple exclusion process. A microscopic shock is a random position X t such that the system as seen from this position at time t has a stationary distribution which is equivalent to the product measure with densities p and A to the left and right of the origin respectively. The diffusion coefficient of the shock D = limt-+oo t-I (E(Xt)2-(EXt )2) has been found to be D = (λ - p)-I (p(1- p) + λ(1 - λ)). In the scale Vi the position of X t is determined by the initial distribution of particles in a region of lenght proportional to t. The distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities p and A. This is the so called dynamical phase transition. Under shock initial conditions the density fluctuation fields depend on the initial configuration. The results are a little weaker in the asymmetric case, when jwnps to the left are also allowed.