Communications in Mathematical Physics

, Volume 183, Issue 3, pp 571–607 | Cite as

Stochastic Burgers and KPZ Equations from Particle Systems

  • Lorenzo Bertini
  • Giambattista Giacomin


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 Ornstein-Uhlenbeck 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 Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.


Burger Equation Scaling Limit Fluctuation Field Macroscopic Equation Diffusive Scaling 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Lorenzo Bertini
    • 1
  • Giambattista Giacomin
    • 2
  1. 1.Centre de Physique Theorique, Ecole Polytechnique, F-91128 Palaiseau Cedex, France. E-mail: bertini@orphee.polytechnique.frFR
  2. 2.Institut für Angewandte Mathematik der Universität Zürich–Irchel, Winterthurerstr. 190, CH-8057 Zürich, Switzerland. E-mail: gbg@amath.unizh.chCH

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