Advertisement

Journal of Statistical Physics

, Volume 117, Issue 1–2, pp 77–98 | Cite as

Multiple Shocks in Bricklayers' Model

  • Márton Balázs
Article

Abstract

In bricklayers' model, which is a generalization of the misanthrope processes, we show that a nontrivial class of product distributions is closed under the time-evolution of the process. This class also includes measures fitting to shock data of the limiting PDE. In particular, we show that shocks of this type with discontinuity of size one perform ordinary nearest neighbor random walks only interacting, in an attractive way, via their jump rates. Our results are related to those of Belitsky and Schütz(4) on the simple exclusion process, although we do not use quantum formalism as they do. The structures we find are described from a fixed position. Similar ones were found in Balázs,(2) valak as seen from the random position of the second class particle.

Multiple shocks shock measure bricklayers' process misanthrope process zero range 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    E.D. Andjel,Invariant measures for the zero range process. The Annals of Prob. 10 (3):325–547(1982).Google Scholar
  2. 2.
    M.Balázs,Microscopic shape of shocks in a domain growth model.J.Stat.Phys. 105(3/4):511–524(2001).Google Scholar
  3. 3.
    M. Balázs,Growth fluctuations in a class of deposition models.Annales de l' Institut Henri Poincaré-PR 39:639–685,2003.Google Scholar
  4. 4.
    V. Belitsky and G. M.Schütz,Diffusion and scattering of shocks in the partially asym-metric simple exclusion process.Electronic Journal of Probability 7 (10):1–12(2002).Google Scholar
  5. 5.
    C. Cocozza-Thivent,--Processus des misanthropes.Z.Wahrsch.Verw.Gebiete 70:509–523 (1985).Google Scholar
  6. 6.
    A. De Masi, C. Kipnis, E. Presutti,and E. Saada,Microscopic structure at the shock in the asymmetric simple exclusion.Stochastics 27:151–165(1988).Google Scholar
  7. 7.
    B. Derrida, J.L. Lebowitz,and E.R. Speer,Shock pro les for the asymmetric simple exclusion process in one dimension.J.Stat.Phys. 89(1-2):135–167(1997).Google Scholar
  8. 8.
    P.A. Ferrari,Shock fluctuations in asymmetric simple exclusion.Probab.Theory Relat. Fields 91:81–102(1992).Google Scholar
  9. 9.
    P.A. Ferrari, L.R.G. Fontes,and Y. Kohayakawa,Invariant measures for a two-species asymmetric process.J.Stat.Phys. 76:1153–1177(1994).Google Scholar
  10. 10.
    P.A. Ferrari, L.R.G. Fontes,and M.E. Vares,The asymmetric simple exclusion model with multiple shocks.Ann.Inst.H.Poincare–PR 36(2):109–126(2000).Google Scholar
  11. 11.
    C. Kipnis and C. Landim,Scaling limits of interacting particle systems. (Springer-Verlag, Berlin,1999).Google Scholar
  12. 12.
    C. Quant,On the construction and stationary distributions of some spatial queueing and particle systems.PhD thesis,Utrecht University,2002.Google Scholar
  13. 13.
    T. Seppäläinen,Existence of hydrodynamics for the totally asymmetric simple K-exclu-sion process.Ann.Probab. 27(1):361–415(1999).Google Scholar
  14. 14.
    J. Smoller,Shock Waves and Reaction-Diffusion Equations.(Springer-Verlag,1983).Google Scholar
  15. 15.
    B. Tóth and B. Valkó,Between equilibrium fluctuations and eulerian scaling:pertur-bation of equilibrium for a class of deposition models.J.Stat.Phys. 109(1/2):177–205 (2002).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • Márton Balázs
    • 1
  1. 1.University of Wisconsin-MadisonUSA; e-mail:

Personalised recommendations