Stochastic Burgers and KPZ Equations from Particle Systems
Abstract
We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized OrnsteinUhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the KardarParisiZhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some nonequilibrium processes. Our main tool is the ColeHopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 February 1997
 DOI:
 10.1007/s002200050044
 Bibcode:
 1997CMaPh.183..571B