Skip to main content
SpringerLink
Log in

Shock profiles for the asymmetric simple exclusion process in one dimension

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at ratesp and 1-p (herep > 1/2) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial and temporal scales by the inviscid Burgers’ equation; the latter has shock solutions with a discontinuous jump from left density ρ- to right density ρ+, ρ-< ρ +, which travel with velocity (2p−1 )(1−ρ+−p −). In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to and travels with the shock. In this paper we obtain the time-invariant measure for this shock solution in the ASEP, as seen from such a particle. The mean density at lattice siten, measured from this particle, approachesp ± at an exponential rate asn→ ±∞, witha characteristic length which becomes independent ofp when\(p/(1 - p) > \sqrt {p_ + (1 - p_ - )/p_ - (1 - p_ + )} \). For a special value of the asymmetry, given byp/(1−p)=p +(1−p −)/p −(1−p +), the measure is Bernoulli, with densityρ − on the left andp + on the right. In the weakly asymmetric limit, 2p−1 → 0, the microscopic width of the shock diverges as (2p+1)-1. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Derrida, S. A. Janowsky, J. L. Lebowitz, and E. R. Speer, Exact solution of the totally asymmetric simple exclusion process: shock profiles,J. Stat. Phys. 73 :813–842 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  2. B. Derrida, S. A. Janowsky, J. L. Lebowitz, and E. R. Speer, Microscopic shock profiles: exact solution of a nonequilibrium system,Europhys. Lett. 22: 651–656 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  3. H. Spohn,Large-Scale Dynamics of Interacting Particles, Texts and Monographs in Physics (Springer-Verlag, New York, 1991). See also references therein.

    Google Scholar 

  4. A. De Masi and E. Presutti,Mathematical Methods for Hydrodynamic Limits, Lecture Notes in Mathematics 1501 (Springer-Verlag, New York, 1991). See also references therein.

    Google Scholar 

  5. J. L. Lebowitz, E. Presutti, and H. Spohn, Microscopic models of hydrodynamic behavior,J. Stat. Phys. 51:841–862 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  6. F. Spitzer, Interaction of Markov processes,Advances in Math. 5:246–290 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Cercignani,Mathematical Methods in Kinetic Theory, 2nd ed. (Plenum Press, New York, 1990).

    MATH  Google Scholar 

  8. T. M. Liggett,Interacting Particle Systems (Springer-Verlag, New York, 1985). See also references therein.

    MATH  Google Scholar 

  9. H. Rost, Nonequilibrium behavior of many particle process: density profiles and local equilibria,Z. Wahrsch. Verw. Gebiete 58 :41–53, (1981).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. E. D. Andjel and M. E. Vares, Hydrodynamical equations for attractive particle systems on Z,J. Stat. Phys. 47:265–288 (1987).

    Article  MathSciNet  Google Scholar 

  11. A. Benassi and J. P. Fouque, “Hydrodynamic limit for the asymmetric simple exclusion process,”Ann. Prob. 15:546–560 (1987). The proof in this reference is incomplete.

    MATH  MathSciNet  Google Scholar 

  12. F. Razakhanlou, Hydrodynamic limit for attractive particle systems on Zd,Commun. Math. Phys. 40:417–448 (1991).

    Article  ADS  Google Scholar 

  13. B. Derrida, S. Goldstein, J. L. Lebowitz, and E. R. Speer, Intrinsic structure of shocks in the asymmetric simple exclusion process, in preparation.

  14. E. D. Andjel, M. Bramson, and T. M. Liggett, Shocks in the asymmetric exclusion process,Probab. Theory Relat. Fields 78:231–247 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Ferrari, Shock fluctuations in asymmetric simple exclusion,Probab. Theory Relat. Fields 91:81–101 (1992).

    Article  MATH  Google Scholar 

  16. F. Razakhanlou, Microscopic structure of shocks in one conservation laws,Ann. Inst. Henri Poincaré: Analyse non linéaire 12 :119–153 (1995).

    Google Scholar 

  17. P. A. Ferrari and C. Kipnis, Second-class particles in the rarefaction fan,Ann. Inst. Henri Poincaré: Probabilités et Statistiques 31 :143–154 (1995).

    MATH  MathSciNet  Google Scholar 

  18. P. Ferrari, C. Kipnis, and E. Saada, Microscopic structure of traveling waves in the asymmetric simple exclusion,Ann. Probab. 19:226–244 (1991).

    MATH  MathSciNet  Google Scholar 

  19. D. Wick, A dynamical phase transition in an infinite particle system,J. Stat. Phys. 38:1015–1025 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Ferrari, The simple exclusion process as seen from a tagged particle,Ann. Probab. 14:1277–1290 (1986).

    MATH  MathSciNet  Google Scholar 

  21. A. De Masi, C. Kipnis, E. Presutti, and E. Saada, Microscopic structure at the shock in the asymmetric simple exclusion,Stoch. and Stoch. Rep. 27:151–165 (1989).

    MATH  Google Scholar 

  22. 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).

    Article  Google Scholar 

  23. P. Ferrari and L. R. G. Fontes, Shock fluctuations in the asymmetric simple exclusion process,Probab. Theory Relat. Fields 99:305–319 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  24. 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).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. S. Sandow, Partially asymmetric exclusion process with open boundaries,Phys. Rev. E 50:2660–2667(1994).

    Article  ADS  Google Scholar 

  26. E. R. Speer, The two species totally asymmetric exclusion process, inOn Two Levels: Micro, Meso and Macroscopic Approaches in Physics, ed. M. Fannes, C. Maes, and A. Verbeure, (Plenum, New York, 1994).

    Google Scholar 

  27. R. B. Stinchcombe and G. M. Schütz, Operator algebra for stochastic dynamics and the Heisenberg chain,Europhys. Lett. 29:663–667 (1995).

    Article  Google Scholar 

  28. R. B. Stinchcombe and G. M. Schütz, Application of operator algebras to stochastic dynamics and the Heisenberg chain,Phys. Rev. Lett. 75:140–143 (1995).

    Article  ADS  Google Scholar 

  29. F. H. L. Essler and V. Rittenberg, Representation of the quadratic algebra and partially asymmetric diffusion with open boundaries,J. Phys. A 29:3375–3407 (1996).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. H. Hinrichsen, S. Sandow, and I. Peschel, On matrix product ground states for reactiondiffusion models,J. Phys. A 29:2643–2649 (1996).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. K. Mallick, Shocks in the asymmetric simple exclusion model with an impurity,J. Phys. A 29:5375–5386(1996).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. M. R. Evans, Bose-Einstein condensation in disordered exclusion models and relation to traffic flow,Europhys. Lett. 36:13–18 (1996).

    Article  ADS  Google Scholar 

  33. B. Derrida and K. Mallick, Exact diffusion constant for the one dimensional partially asymmetric exclusion model,J. Phys. A 30:1031–1046 (1997).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. B. Derrida and M. R. Evans, The asymmetric exclusion model: exact results through a matrix approach, inNonequilibrium Statistical Mechanics in One Dimension, ed. V. Privman (Cambridge University Press, Cambridge, 1996).

    Google Scholar 

  35. A. Honecker and I. Peschel, Matrix-product states for a one-dimensional lattice gas with parallel dynamics, preprint 1996.

  36. A. De Masi, E. Presutti, and E. Scacciatelli, The weakly asymmetric simple exclusion process,Ann. Inst. Henri Poincaré: Probabilités et Statistiques 25 :1–38 (1989).

    MATH  Google Scholar 

  37. C. Kipnis, S. Olla, and S. R. S. Varadhan, Hydrodynamics and large deviation for simple exclusion processes,Commun. Pure App. Math. 42:115–137 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  38. J. L. Lebowitz, E. Orlandi, and E. Presutti, Convergence of stochastic cellular automaton to Burgers’ equation: fluctuations and stability,Physica D 33 :165–188 (1988).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. J. Stephenson, Ising model spin correlations on the triangular lattice. IV. Anisotropic ferromagnetic and antiferromagnetic lattices,J. Math. Phys. 11:420–431 (1970).

    Article  ADS  MathSciNet  Google Scholar 

  40. P. Rujàn: Cellular automata and statistical mechanics models,J. Stat. Phys. 49 :139–222 (1987).

    Article  MATH  Google Scholar 

  41. D. Kim, Bethe Ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar-Parisi-Zhang growth model,Phys. Rev. E 52:3512–3524 (1995).

    Article  ADS  Google Scholar 

  42. J. Neergard and M. den Nijs, Crossover scaling functions in one dimensional growth models,Phys. Rev. Lett. 74:730–733 (1995).

    Article  ADS  Google Scholar 

  43. K. Mallick and S. Sandow, Finite dimensional representations of the quadratic algebra: applications to the exclusion process,J. Phys. A 30 :4513–4526 (1997).

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Statistique, Ecole Normale Supérieure, 75005, Paris, France

    B. Derrida

  2. Institut des Hautes Etudes Scientifiques, F-91440, Bures-sur-Yvette, France

    J. L. Lebowitz & E. R. Speer

  3. Department of Mathematics, Rutgers University, 08903, New Brunswick, New Jersey

    E. R. Speer

Authors
  1. B. Derrida
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. L. Lebowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. R. Speer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Derrida.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Derrida, B., Lebowitz, J.L. & Speer, E.R. Shock profiles for the asymmetric simple exclusion process in one dimension. J Stat Phys 89, 135–167 (1997). https://doi.org/10.1007/BF02770758

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02770758

Key Words

Search

Navigation