MICROSCOPIC STRUCTURE OF TRAVELING WAVES IN THE ASYMMETRIC SIMPLE EXCLUSION PROCESS

The one-dimensional nearest neighbor asymmetric simple exclusion process has been used as a microscopic approximation for the Burgers equation. This equation has travelling wave solutions. In this paper we show that those solutions have a microscopic structure. More precisely, we consider the simple exclusion process with rate p (respectively, q = 1 - p) for jumps to the right (left), 1/2 < p less-than-or-equal-to 1, and we prove the following results: There exists a measure mu on the space of configurations approaching asymptotically the product measure with densities rho and lambda to the left and right of the origin, respectively, rho < lambda, and there exists a random position X(t) membership-sign Z, such that, at time t, the system "as seen from X(t)," remains distributed according to mu, for all t greater-than-or-equal-to 0. The hydrodynamical limit for the simple exclusion process with initial measure mu coverges to the travelling wave solution of the inviscid Burgers equation. The random position X(t)/t converges strongly to the speed v = (1 - lambda - p)(p - q) of the travelling wave. Finally, in the weakly asymmetric hydrodynamical limit, the stationary density profile converges to the travelling wave solution of the Burgers equation.