Journal of Statistical Physics

, Volume 73, Issue 5–6, pp 813–842 | Cite as

Exact solution of the totally asymmetric simple exclusion process: Shock profiles

  • B. Derrida
  • S. A. Janowsky
  • J. L. Lebowitz
  • E. R. Speer
Articles

Abstract

The microscopic structure of macroscopic shocks in the one-dimensional, totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles-“first” and “second” class. This nonequilibrium steady state factorizes about any second-class particle, which implies factorization in the one-component system about the (random) shock position. It also exhibits several other interesting features, including long-range correlations in the limit of zero density of the second-class particles. The solution also shows that a finite number of second-class particles in a uniform background of first-class particles form a weakly bound state.

Key words

Asymmetric simple exclusion process shock profiles secondclass particles Burgers equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Spohn,Large-Scale Dynamics of Interacting Particles (Springer-Verlag, New York, 1991); A. De Masi and E. Presutti,Mathematical Methods for Hydrodynamic Limits (Springer-Verlag, New York, 1991); and references therein.Google Scholar
  2. 2.
    J. Lebowitz, E. Presutti, and H. Spohn, Microscopic models of hydrodynamic behavior,J. Stat. Phys. 51:841–862 (1988).Google Scholar
  3. 3.
    B. Schmittman, Critical behavior of the driven diffusive lattice gas,Int. J. Mod. Phys. B 4:2269–2306 (1990).Google Scholar
  4. 4.
    P. Garrido, J. Lebowitz, C. Maes, and H. Spohn, Long-range correlations for conservative dynamics,Phys. Rev. A 42:1954–1968 (1990).Google Scholar
  5. 5.
    R. Bhagavatula, G. Grinstein, Y. He, and C. Jayaprakash, Algebraic correlations in conserving chaotic systems,Phys. Rev. Lett. 69:3483–3486 (1992).Google Scholar
  6. 6.
    T. M. Liggett,Interacting Particle Systems (Springer-Verlag, New York, 1985).Google Scholar
  7. 6a.
    See also T. M. Liggett, Ergodic theorems for the asymmetric simple exclusion process,Trans. Amer. Math. Soc. 213, 237–261 (1976).Google Scholar
  8. 6b.
    T. M. Liggett, Ergodic theorems for the asymmetric simple exclusion process II,Ann. Prob. 5, 795–801 (1977).Google Scholar
  9. 7.
    H. Rost, Nonequilibrium behavior of many particle process: Density profiles and local equilibria,Z. Wahrsch. Verw. Gebiete 58:41–53 (1981).Google Scholar
  10. 8.
    A. Benassi and J. P. Fouque, Hydrodynamic limit for the asymmetric simple exclusion process,Ann. Prob. 15:546–560, and erratum. (1987).Google Scholar
  11. 9.
    E. D. Andjel and M. E. Vares, Hydrodynamical equations for attractive particle systems on ℤ,J. Stat. Phys. 47:265–288 (1987).Google Scholar
  12. 10.
    D. Wick, A dynamical phase transition in an infinite particle system,J. Stat. Phys. 38:1015–1025 (1985).Google Scholar
  13. 11.
    P. Ferrari, The simple exclusion process as seen from a tagged particle,Ann. Prob. 14:1277–1290 (1986).Google Scholar
  14. 12.
    E. D. Andjel, M. Bramson, and T. M. Liggett, Shocks in the asymmetric exclusion process,Prob. Theory Related Fields 78:231–247 (1988).Google Scholar
  15. 13.
    A. De Masi, C. Kipnis, E. Presutti, and E. Saada, Microscopic structure at the shock in the asymmetric simple exclusion,Stoch. Stoch. Rep. 27:151–165 (1989).Google Scholar
  16. 14.
    P. Ferrari, C. Kipnis, and E. Saada, Microscopic structure of traveling waves in the asymmetric simple exclusion,Ann. Prob. 19:226–244 (1991).Google Scholar
  17. 15.
    P. Ferrari, Shock fluctuations in asymmetric simple exclusion,Prob. Theory Related Fields 91:81–101 (1992).Google Scholar
  18. 15a.
    See also P. A. Ferrari and L. R. G. Fontes, Shock fluctuations in the asymmetric simple exclusion process (1993), to appear inProb. Theory Related Fields.Google Scholar
  19. 16.
    C. Boldrighini, G. Cosimi, S. Frigio, and M. G. Nuñes, Computer simulation of shock waves in the completely asymmetric simple exclusion process,J. Stat. Phys. 55:611–623 (1989).Google Scholar
  20. 17.
    S. A. Janowsky and J. L. Lebowitz, Finite size effects and shock fluctuations in the asymmetric simple exclusion process,Phys. Rev. A 45:618–625 (1992).Google Scholar
  21. 18.
    B. Derrida, S. A. Janowsky, J. L. Lebowitz, and E. R. Speer, Microscopic shock profiles: Exact solution of a nonequilibrium system,Europhys. Lett. 22 (1993).Google Scholar
  22. 19.
    B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, An exact solution of a 1D asymmetric exclusion model using a matrix formulation,J. Phys. A 26:1493–1517 (1993).Google Scholar
  23. 20.
    B. Derrida, E. Domany, and D. Mukamel, An exact solution of a one dimensional asymmetric exclusion model with open boundaries,J. Stat. Phys. 69:667–687 (1992).Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • B. Derrida
    • 1
  • S. A. Janowsky
    • 2
  • J. L. Lebowitz
    • 4
  • E. R. Speer
    • 2
  1. 1.Service de Physique ThéoriqueCEN SaclayGif-sur-YvetteFrance
  2. 2.Department of MathematicsRutgers UniversityNew Brunswick
  3. 3.Department of MathematicsUniversity of TexasAustin
  4. 4.Departments of Mathematics and PhysicsRutgers UniversityNew Brunswick

Personalised recommendations