Probability Theory and Related Fields

, Volume 99, Issue 2, pp 305–319 | Cite as

Shock fluctuations in the asymmetric simple exclusion process

  • P. A. Ferrari
  • L. R. G. Fontes
Article

Summary

We consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the initial measure isvρ,λ, a product measure with densities ρ and λ to the left and right of the origin respectively, ρ<λ, then there exists a (microscopic) shock for the system. A shock is a random positionXt such that the system as seen from this position at timet has asymptotic product distributions with densities ρ and λ to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=limt→∞t−1(E(Xt)2−(EXt)2) and findD=(p−q)(λ−ρ)−1(ρ(1−ρ)+λ(1−λ)) as conjectured by Spohn (1991). We show that in the scale\(\sqrt t\) the position ofXt is determined by the initial distribution of particles in a region of length proportional tot. 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 ρ and λ. This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.

Mathematics Subject Classification 1991

60K35 82C22 82C24 82C41 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [abl] Andjel, E. D., Bramson, M., Liggett, T. M.: Shocks in the asymmetric simple exclusion process. Probab. Theory Relat. Fields78, 231–247 (1988)Google Scholar
  2. [av] Andjel, E. D., Vares, M. E.: Hydrodynamic equations for attractive particle systems on #x2124. J. Stat. Phys.47, 265–288 (1987)Google Scholar
  3. [bf1] Benassi, A., Fouque, J.-P.: Hydrodynamical limit for the asymmetric simple exclusion process. Ann. Probab.15, 546–560 (1987)Google Scholar
  4. [bf2] Benassi, A., Fouque, J.-P.: Fluctuation field for the asymmetric simple exclusion process. Proceedings of Oberwolfach Conference in SPDE, Nov 89, Basel, Boston, Berlin: BirkhauserGoogle Scholar
  5. [bfsv] Benassi, A., Fouque, J.-P., Saada, E., Vares, M. E.: Asymmetric attractive particle systems on Z: hydrodynamical limit for monotone initial profiles. J. Stat. Phys.63, 719–735 (1991)Google Scholar
  6. [bcfg] 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
  7. [df] De Masi, A., Ferrari, P. A.: Self diffusion in one dimensional lattice gases in the presence of an external field. J. Stat. Phys.38, 603–613 (1985)Google Scholar
  8. [dfv] De Masi, A., Ferrari, P. A., Vares, M. E.: A microscopic model of interface related to the Burgers equation. J. Stat. Phys.55, 3/4, 601–609 (1989)Google Scholar
  9. [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
  10. [f1] Ferrari, P. A.: The simple exclusion process as seen from a tagged particle. Ann. Probab.14, 1277–1290 (1986)Google Scholar
  11. [f2] Ferrari, P. A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields.91, 81–101 (1992)Google Scholar
  12. [ff] Ferrari, P. A., Fontes, L. R. G.: Current fluctuations for the asymmetric simple exclusion process. Ann. Probab.22 (1), (1994)Google Scholar
  13. [fks] 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
  14. [gp] Gärtner, J., Presutti, E.: Shock fluctuations in a particle system. Ann. Inst. Henri Poincaré, Probab. Stat.53, 1–14 (1989)Google Scholar
  15. [k] Kipnis, C.: Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab.14, 397–408 (1986)Google Scholar
  16. [lal] Landim, C.: Conservation of local equilibrium for attractive particle systems onZ d. Ann. Probab. (to appear)Google Scholar
  17. [lax] Lax, P. D.: The formation and decay of shock waves. Am. Math. Mon.79, 227–241 (1972)Google Scholar
  18. Liggett, T. M.: Ergodic theorems for the asymmetric simple exclusion process. Trans. Am. Math. Soc.213, 237–261 (1975)Google Scholar
  19. Liggett, T. M.: Ergodic theorems for the asymmetric simple exclusion process, II. Ann. Probab.4, 339–356 (1977)Google Scholar
  20. Liggett, T. M.: Coupling the simple exclusion process. Ann. Probab.4, 339–356 (1976)Google Scholar
  21. [L] Liggett, T. M.: Interacting Particle Systems. Berlin, Heidelberg, New York (1985): SpringerGoogle Scholar
  22. [re] Rezakhanlou, H.: (1990) Hydrodynamic limit for attractive particle systems onZ d. Commun. Math. Phys.140, 417–448 (1990)Google Scholar
  23. [r] Rost, H.: Nonequilibrium behavior of a many particle process: density profile and local equilibrium. Z. Wahrsch. verw. Gebiete,58, 41–53 (1982)Google Scholar
  24. [s] Spitzer, F.: Interaction of Markov processes. Adv. Math.5, 246–290 (1970)Google Scholar
  25. [S] Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin, Heidelberg, New York: Springer (1991)Google Scholar
  26. [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 1994

Authors and Affiliations

  • P. A. Ferrari
    • 1
  • L. R. G. Fontes
    • 1
  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrasil

Personalised recommendations