Journal of Statistical Physics

, Volume 86, Issue 5–6, pp 1265–1287 | Cite as

Duality relations for asymmetric exclusion processes

  • Gunter M. Schütz


We derive duality relations for a class ofU q [SU(2)]-symmetric stochastic processes, including among others the asymmetric exclusion process in one dimension. Like the known duality relations for symmetric hopping processes, these relations express certainm-point correlation functions inN-particle systems (N≥m) in terms of sums of correlation functions of the same system but with onlym particles. For the totally asymmetric case we obtain exact expressions for some boundary density correlation functions. The dynamical exponent for these correlators isz=2, which is different from the dynamical exponent for bulk density correlations, which is known to bez=3/2.

Key Words

Asymmetric exclusion processes duality relations quantum algebra correlation functions dynamical scaling 


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  1. 1.
    T. Liggett,Interacting Particle Systems (Springer-Verlag, New York, 1985).Google Scholar
  2. 2.
    F. Spitzer,Adv. Math. 5:246 (1970).Google Scholar
  3. 3.
    S. Alexander and T. Holstein,Phys. Rev. B 18:301 (1978).Google Scholar
  4. 4.
    L.-H. Gwa and H. Spohn,Phys. Rev. A 46:844 (1992).Google Scholar
  5. 5.
    F. C. Alcaraz and V. Rittenberg,Phys. Lett. B 314:377 (1993).Google Scholar
  6. 6.
    F. C. Alcaraz, M. Droz, M. Henkel, and V. Rittenberg,Ann. Phys. (N.Y.)230:250 (1994).Google Scholar
  7. 7.
    S. Sandow and G. Schütz,Europhys. Lett. 26:7 (1994).Google Scholar
  8. 8.
    S. R. Dahmen,J. Phys. A. 28:905 (1995).Google Scholar
  9. 9.
    D. Kim,Phys. Rev. E 52:3512 (1995).Google Scholar
  10. 10.
    D. Kandel, E. Domany, and B. Nienhuis,J. Phys. A 23: L755 (1990).Google Scholar
  11. 11.
    G. Schütz,J. Stat. Phys. 71:471 (1993).Google Scholar
  12. 12.
    G. Schütz,Phys. Rev. E 47:4265 (1993).Google Scholar
  13. 13.
    H. Hinrichsen,J. Phys. A 29:3659 (1996).Google Scholar
  14. 14.
    A. Honecker and I. Peschel, Preprint (1996).Google Scholar
  15. 15.
    G. Schütz and S. Sandow,Phys. Rev. E 49:2726 (1994).Google Scholar
  16. 16.
    S. Albeverio and S.-M. Fei, Preprint (1996).Google Scholar
  17. 17.
    J. Krug and H. Spohn, inSolids far from Equilibrium, C. Godreche, ed. (Cambridge University Press, Cambridge, 1991), and references therein.Google Scholar
  18. 18.
    P. Meakin, P. Ramanlal, L. Sander, and R. Ball,Phys. Rev. A 34:5091 (1986).Google Scholar
  19. 19.
    M. Plischke, Z. Racz, and D. Liu,Phys. Rev. B 35:3485 (1987).Google Scholar
  20. 20.
    M. Kardar, G. Parisi, and Y. C. Zhang,Phys. Rev. Lett. 56:889 (1986).Google Scholar
  21. 21.
    J. Krug,Phys. Rev. Lett. 67:1882 (1991).Google Scholar
  22. 22.
    S. A. Janowsky and J. L. Lebowitz,Phys. Rev. A 45:618 (1992).Google Scholar
  23. 23.
    G. Schütz and E. Domany,J. Stat. Phys. 72:277 (1993).Google Scholar
  24. 24.
    B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier,J. Phys. A 26:193 (1993).Google Scholar
  25. 25.
    B. Derrida, S. A. Janowsky, J. L. Lebowitz, and E. R. Speer,Europhys. Lett. 22: 651 (1993).Google Scholar
  26. 26.
    M. Kardar and Y. C. Zhang,Phys. Rev. Lett. 58:2087 (1987).Google Scholar
  27. 27.
    D. S. Fisher and D. A. Huse,Phys. Rev. B 43:10728 (1991).Google Scholar
  28. 28.
    L.-H. Tang and I. F. Lyuksyutov,Phys. Rev. Lett. 71:2745 (1993).Google Scholar
  29. 29.
    J. Krug and L.-H. Tang,Phys. Rev. E 50:104 (1994).Google Scholar
  30. 30.
    A. Schadschneider and M. Schreckenberg,J. Phys. A 26:L679 (1993).Google Scholar
  31. 31.
    K. Nagel and M. Schreckenberg,J. Phys. (Paris)I 2:2221 (1993).Google Scholar
  32. 32.
    K. Nagel,Phys. Rev. E 53:4655 (1996).Google Scholar
  33. 33.
    A. N. Kirrilov and N. Yu. Reshetikhin, LOMI preprint (1988).Google Scholar
  34. 34.
    V. Pasquier and H. Saleur,Nucl. Phys. B 330:523 (1990).Google Scholar
  35. 35.
    J. Fuchs,Affine Lie-Algebras and Quantum Groups (Cambridge University Press, Cambridge, 1992).Google Scholar
  36. 36.
    E. Hopf,Commun. Pure Appl. Math. 3:201 (1950); J. D. Cole,Q. Appl. Math. 9:225 (1951).Google Scholar
  37. 37.
    U. Schultz, J. Villain, E. Brézin, and H. Orland,J. Stat. Phys. 51:1 (1988).Google Scholar
  38. 38.
    G. Forgacs, R. Lipowsky, and Th. M. Nieuwenhuizen, inPhase Transitions and Critical Phenomena, Vol. 14, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1991), and references therein.Google Scholar
  39. 39.
    M. Doi,J. Phys. A 9:1465, 1479 (1976).Google Scholar
  40. 40.
    P. Grassberger and M. Scheunert,Fortschr. Phys. 28:547 (1980).Google Scholar
  41. 41.
    S. Sandow and S. Trimper,Europhys. Lett. 21:799 (1993).Google Scholar
  42. 42.
    R. B. Stinchcombe, M. D. Grynberg, and M. Barma,Phys. Rev. E 47:1018 (1993).Google Scholar
  43. 43.
    H. N. V. Temperley and E. H. Lieb,Proc. R. Soc. A 322:25 (1971).Google Scholar
  44. 44.
    M. T. Batchelor, L. Mezincescu, R. I. Nepomechie, and V. Rittenberg,J. Phys. A 23 L141 (1990).Google Scholar
  45. 45.
    H. J. Vega and E. Lopes,Nucl. Phys. B 362:261 (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Gunter M. Schütz
    • 1
    • 2
  1. 1.Isaac Newton Institute of Mathematical SciencesUniversity of CambridgeCambridgeUK
  2. 2.Department of PhysicsUniversity of Oxford, Theoretical PhysicsOxfordUK

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