SHOCK FLUCTUATIONS IN THE ASYMMETRIC SIMPLE EXCLUSION PROCESS

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 et al. (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, 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-->infinity) 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 square-root t the position of X(t) is determined by the initial distribution of particles in a region of length 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.