Pa. Ferrari et Lrg. Fontes, SHOCK FLUCTUATIONS IN THE ASYMMETRIC SIMPLE EXCLUSION PROCESS, Probability theory and related fields, 99(2), 1994, pp. 305-319
We consider the one dimensional nearest neighbors asymmetric simple ex
clusion process with rates q and p for left and right jumps respective
ly; q < p. Ferrari et al. (1991) have shown that if the initial measur
e is nu(rho, lambda), a product measure with densities rho and lambda
to the left and right of the origin respectively, rho < lambda, then t
here 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 comp
ute 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 init
ial 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 trans
ition. Under shock initial conditions we show how the density fluctuat
ion fields depend on the initial configuration.