Probability Theory and Related Fields

, Volume 91, Issue 1, pp 81–101 | Cite as

Shock fluctuations in asymmetric simple exclusion

  • Pablo A. Ferrari


The one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumpsp>q to the right and left, respectively, and with initial product measure with densities ϱ<λ to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density (ϱ=0) we prove, in the scale t1/2, that the position of the shock at timet depends only on the initial configuration in a region depending ont. The proofs are based on laws of large numbers for the second class particle.

AMS 1980 Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ab] Andjel, E.D., Bramson, M., Liggett, T.M.: Shoeks in the asymmetric simple exclusion process. Probab. Theory Relat. Fields78, 231–247 (1988)Google Scholar
  2. [ak] Andjel, E.D., Kipnis, C.: Derivation of the hydrodynamical equations for the zero-range interaction process: a nonlinear Euler equation. Ann. Probab.12, 325–334 (1984)Google Scholar
  3. [av] Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on ℤ J. Stat. Phys.47, 265–288 (1987)Google Scholar
  4. [befg] Boldrighini, C., Cosimi, C., Frigio, A., Grasso-Nunes, M.: Computer simulations of shock waves in completely asymmetric simple exclusion process. J. Stat. Phys.55, 611–623 (1989)Google Scholar
  5. [b] Bramson, M.: Front propagation in certain one dimensional exelusion models. J. Stat. Phys.51, 863–869 (1988)Google Scholar
  6. [df] De Masi, A., Ferrari, P.A.: Self difusion in one dimensional lattice gases in the presence of an external field. J. Stat. Phys.38, 603–613 (1985)Google Scholar
  7. [dkps] De Masi, A., Kipnis, C., Presutti, E., Saada, E.: Microscopic structure at the shock in the asymmetric simple exclusion. Stochastics27, 151–165 (1988)Google Scholar
  8. [f] Ferrari, P.A.: The simple exclusion process as seen from a tagged particle. Ann. Probab.14, 1277–1290 (1986)Google Scholar
  9. [lks] Ferrari, P.A., Kipnis, C., Saada, E.: Microscopic structure of travelling waves for asymmetric simple exclusion process. Ann. Probab.19, 226–244 (1991).Google Scholar
  10. [gp] Gärtner, J., Presutti, E.: Shock fluctuations in a particle system. Ann. Inst. Henri Poincare B53, 1–14 (1990)Google Scholar
  11. [k] Kipnis, C.: Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab.14, 397–408 (1986)Google Scholar
  12. [lps] Lebowitz, J.L., Presutti, E., Spohn, H.: Microscopic models of hydrodynamical behavior. J. Stat. Phys.51, 841–862 (1988)Google Scholar
  13. [l] Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab.4, 339–356 (1976)Google Scholar
  14. [L] Liggett, T.M.: Interacting particle systems. Berlin Heidelberg New York: Springer 1985Google Scholar
  15. [s] Saada, E.: A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab.15, 375–381 (1987)Google Scholar
  16. [S] Spohn, H.: Large scale dynamics of interacting particles. Part B: Stochastic lattice gases. (Preprint, 1989)Google Scholar
  17. [vb] Van Beijeren, H.: Fluctuations in the motions of mass and of patterns in one-dimensional driven diffusive systems. J. Stat. Phys.63, 47–58 (1991)Google Scholar
  18. [w] Wick, D.: A dynamical phase transition in an infinite particle system. J. Stat. Phys.38, 1015–1025 (1985)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Pablo A. Ferrari
    • 1
  1. 1.Instituto de Matemática e EstatisticaUniversidade de São PauloPauloBrasil

Personalised recommendations