Microscopic structure of travelling waves in the asymmetric simple exclusion process. (English) Zbl 0725.60113

Ferrari, P. A.; Kipnis, C.; Saada, E.

Ann. Probab. 19, No. 1, 226-244 (1991).

Consider the one-dimensional simple exclusion process with rate \(p\in (1/2,1]\) (resp., \(q=1-p)\) for jump to the right (left). The authors prove the following results: There exists a measure \(\mu\) on the space of configurations approaching asymptotically the product measure with densities \(\rho\) and \(\lambda\) \((\rho <\lambda)\) respectively to the left and right of the origin. There exists a random position X(t)\(\in {\mathbb{Z}}\) such that the system “as seen from X(t)” at time t is distributed according to \(\mu\) for all \(t\geq 0\). Next, the hydrodynamical limit for the process with initial distribution \(\mu\) converges to the travelling wave solution of the inviscid Burgers equation. Moreover, X(t)/t converges strongly to the speed \(v=(1-\lambda -\rho)(p-q)\) 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. These results show that the travelling wave solutions to the Burgers equation have a microscopic structure.

Reviewer: Chen Mu-fa (Beijing)

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

exclusion process; hydrodynamical limit; weakly asymmetric hydrodynamical limit; travelling wave solutions; Burgers equation