Journal of Statistical Physics

, Volume 153, Issue 3, pp 377–399 | Cite as

Coupled Kardar-Parisi-Zhang Equations in One Dimension

  • Patrik L. Ferrari
  • Tomohiro Sasamoto
  • Herbert Spohn


Over the past years our understanding of the scaling properties of the solutions to the one-dimensional KPZ equation has advanced considerably, both theoretically and experimentally. In our contribution we export these insights to the case of coupled KPZ equations in one dimension. We establish equivalence with nonlinear fluctuating hydrodynamics for multi-component driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the two-component AHR model. Its steady state is computed using the matrix product ansatz. Thereby all coefficients appearing in the coupled KPZ equations are deduced from the microscopic model. Time correlations in the steady state are simulated and we confirm not only the scaling exponent, but also the scaling function and the non-universal coefficients.


KPZ equation Universality Scaling functions Interacting particle system Exclusion processes Matrix product 



We thank H. van Beijeren and G. Schütz for most informative discussions and J. Krug for pointing out the early literature on coupled KPZ equations. P.L. Ferrari is grateful for the hospitality at the TU-Munich, where part of the work was made. His work is supported by the German Research Foundation via the SFB 1060–B04 project.


  1. 1.
    Alcaraz, F.C., Droz, M., Henkel, M., Rittenberg, V.: Reaction-diffusion processes, critical dynamics, and quantum chains. Ann. Phys. 230, 250–302 (1994) MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Alves, S.G., Oliveira, T.J., Ferreira, S.C.: Universal fluctuations in radial growth models belonging to the KPZ universality class. Europhys. Lett. 96, 48003 (2011) ADSCrossRefGoogle Scholar
  3. 3.
    Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. Math. 64, 466–537 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arita, C., Kuniba, A., Sakai, K., Sawabe, T.: Spectrum of a multi-species asymmetric simple exclusion process on a ring. J. Phys. A, Math. Theor. 42, 345002 (2009) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Arndt, P.F., Heinzel, T., Rittenberg, V.: Spontaneous breaking of translational invariance in one-dimensional stationary states on a ring. J. Phys. A 31, L45 (1998) MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Arndt, P.F., Heinzel, T., Rittenberg, V.: Spontaneous breaking of translational invariance and spatial condensation in stationary states on a ring. I. The neutral system. J. Stat. Phys. 97, 1–65 (1999) MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Baik, J., Ferrari, P.L., Péché, S.: Convergence of the two-point function of the stationary TASEP. In: Griebel, M. (ed.) Singular Phenomena and Scaling in Mathematical Models. Springer, Berlin (2013) Google Scholar
  8. 8.
    Basu, A., Bhattacharjee, J.K., Ramaswamy, S.: Mean magnetic field and noise cross-correlation in magnetohydrodynamic turbulence: results from a one-dimensional model. Eur. Phys. J. B 9, 725–730 (1999) ADSCrossRefGoogle Scholar
  9. 9.
    Bernardin, C., Gonçalves, P.: Anomalous fluctuations for a perturbed Hamiltonian system with exponential interactions. Commun. Math. Phys. (2013) (to appear). arXiv:1205.1879
  10. 10.
    Bernardin, C., Olla, S.: Transport properties of a chain of anharmonic oscillators with random flip of velocities. J. Stat. Phys. 145, 1224–1255 (2011) MathSciNetADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Bernardin, C., Stoltz, G.: Anomalous diffusion for a class of systems with two conserved quantities. Nonlinearity 25, 1099 (2012) MathSciNetADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle system. Commun. Math. Phys. 183, 571–607 (1997) MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields (2013) (online first) Google Scholar
  14. 14.
    Chen, S., Zhang, Y., Wang, J., Zhao, H.: Diffusion of heat, energy, momentum, and mass in one-dimensional systems. Phys. Rev. E 87, 032153 (2013) ADSCrossRefGoogle Scholar
  15. 15.
    Das, D., Basu, A., Barma, M., Ramaswamy, S.: Weak and strong dynamic scaling in a one-dimensional driven coupled-field model: effects of kinematic waves. Phys. Rev. E 64, 021402 (2001) ADSCrossRefGoogle Scholar
  16. 16.
    Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution of a 1D exclusion model using a matrix formulation. J. Phys. A 26, 1493–1517 (1993) MathSciNetADSCrossRefzbMATHGoogle Scholar
  17. 17.
    Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57, 457–537 (2008) ADSCrossRefGoogle Scholar
  18. 18.
    Ertaş, D., Kardar, M.: Dynamic roughening of directed lines. Phys. Rev. Lett. 69, 929–932 (1992) ADSCrossRefGoogle Scholar
  19. 19.
    Ertaş, D., Kardar, M.: Dynamic relaxation of drifting polymers: a phenomenological approach. Phys. Rev. E 48, 1228–1245 (1993) ADSCrossRefGoogle Scholar
  20. 20.
    Esposito, R., Marra, R., Yau, H.T.: Diffusive limit of asymmetric simple exclusion. Rev. Math. Phys. 06, 1233–1267 (1994) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ferrari, P.L., Frings, R.: Finite time corrections in KPZ growth models. J. Stat. Phys. 144, 1123–1150 (2011) MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Ferrari, P.L., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Commun. Math. Phys. 265, 1–44 (2006) MathSciNetADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Fleischer, J., Diamond, P.H.: Compressible Alfven turbulence in one dimension. Phys. Rev. E 58, R2709–R2712 (1998) MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Funaki, T., Quastel, J.: Invariant measures for a linear stochastic heat equation related to the KPZ equation, Talk at the workshop Entropy and Nonequilibrium Dynamics, Budapest, 23–25 May 2013 Google Scholar
  25. 25.
    Georgii, H.O.: Canonical Gibbs measures: some extensions of de Finetti’s representation theorem for interacting particle systems. In: Lecture Notes in Mathematics, vol. 760. Springer, Berlin (1979) Google Scholar
  26. 26.
    Grisi, R., Schütz, G.M.: Current symmetries for particle systems with several conservation laws. J. Stat. Phys. 145, 1499–1512 (2011) MathSciNetADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178, 559–664 (2013) CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Halpin-Healy, T.: 2+1-dimensional directed polymer in a random medium: scaling phenomena and universal distributions. Phys. Rev. Lett. 109, 170602 (2012) ADSCrossRefGoogle Scholar
  29. 29.
    Imamura, T., Sasamoto, T.: Exact solution for the stationary KPZ equation. Phys. Rev. Lett. 108, 190693 (2012) CrossRefGoogle Scholar
  30. 30.
    Imamura, T., Sasamoto, T.: Stationary correlations for the 1D KPZ equation. J. Stat. Phys. 150, 908–939 (2013) MathSciNetADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Kardar, M.: Nonequilibrium dynamics of interfaces and lines. Phys. Rep. 301, 85–112 (1998) ADSCrossRefGoogle Scholar
  32. 32.
    Kardar, M., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Kim, K.H., den Nijs, M.: Dynamic screening in a two-species asymmetric exclusion process. Phys. Rev. E 76, 21107 (2007) MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Künsch, H.: Non reversible stationary measures for infinite interacting particle systems. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 407–424 (1984) CrossRefzbMATHGoogle Scholar
  35. 35.
    Lahiri, R., Ramaswamy, S.: Are steadily moving crystals unstable? Phys. Rev. Lett. 79, 1150–1153 (1997) ADSCrossRefGoogle Scholar
  36. 36.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003) MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Levine, A., Ramaswamy, S., Frey, E., Bruinsma, R.: Screened and unscreened phases in sedimenting suspensions. Phys. Rev. Lett. 81, 5944–5947 (1998) ADSCrossRefGoogle Scholar
  38. 38.
    Mendl, C.B., Spohn, H.: Dynamic correlators of FPU chains and nonlinear fluctuating hydrodynamics (2013). arXiv:1305.1209
  39. 39.
    Miettinen, L., Myllys, M., Merikoski, J., Timonen, J.: Experimental determination of KPZ height-fluctuation distributions. Eur. Phys. J. B 46, 55–60 (2005) ADSCrossRefGoogle Scholar
  40. 40.
    Popkov, V., Schütz, G.M.: Unusual shock wave in two-species driven systems with an umbilic point. Phys. Rev. E 86, 031139 (2012) ADSCrossRefGoogle Scholar
  41. 41.
    Popkov, V., Fouladvand, M.E., Schütz, G.M.: A sufficient criterion for integrability of stochastic many-body dynamics and quantum spin chains. J. Phys. A, Math. Gen. 35, 7187–7204 (2002) ADSCrossRefzbMATHGoogle Scholar
  42. 42.
    Prähofer, M.: Exact scaling function for one-dimensional stationary KPZ growth (2002).
  43. 43.
    Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: Sidoravicius, V. (ed.) In and Out of Equilibrium. Progress in Probability. Birkhäuser, Basel (2002) Google Scholar
  44. 44.
    Sasamoto, T.: One-dimensional partially asymmetric simple exclusion process on a ring with a defect particle. Phys. Rev. E 61, 4980–4990 (2000) MathSciNetADSCrossRefGoogle Scholar
  45. 45.
    Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834, 523–542 (2010) MathSciNetADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Sasamoto, T., Rajewsky, N., Speer, E.R.: Spatial particle condensation for an exclusion process on a ring. Physica A 279, 123–142 (2000) ADSCrossRefGoogle Scholar
  47. 47.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer, Heidelberg (1991) CrossRefzbMATHGoogle Scholar
  48. 48.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains (2013). arXiv:1305.6412
  49. 49.
    Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice Eden model. J. Stat. Mech., P05007 (2012) Google Scholar
  50. 50.
    Takeuchi, K.A., Sano, M.: Growing interfaces of liquid crystal turbulence: universal scaling and fluctuations. Phys. Rev. Lett. 104, 230601 (2010) ADSCrossRefGoogle Scholar
  51. 51.
    Takeuchi, K.A., Sano, M.: Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence. J. Stat. Phys. 147, 853–890 (2012) ADSCrossRefzbMATHGoogle Scholar
  52. 52.
    Tóth, B., Valkó, B.: Onsager relations and Eulerian hydrodynamic limit for systems with several conservation laws. J. Stat. Phys. 112, 497–521 (2003) CrossRefzbMATHGoogle Scholar
  53. 53.
    van Beijeren, H.: Exact results for anomalous transport in one-dimensional Hamiltonian systems. Phys. Rev. Lett. 108, 180601 (2012) CrossRefGoogle Scholar
  54. 54.
    van Beijeren, H., Kutner, R., Spohn, H.: Excess noise for driven diffusive systems. Phys. Rev. Lett. 54, 2026–2029 (1985) MathSciNetADSCrossRefGoogle Scholar
  55. 55.
    Wehefritz-Kaufmann, B.: Dynamical critical exponent for two-species totally asymmetric diffusion on a ring. SIGMA 6, 039 (2010) MathSciNetGoogle Scholar
  56. 56.
    Yanase, S.: New one-dimensional model equations of magnetohydrodynamic turbulence. Phys. Plasmas 4, 1010 (1997) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Patrik L. Ferrari
    • 1
  • Tomohiro Sasamoto
    • 2
    • 3
  • Herbert Spohn
    • 3
  1. 1.Institute for Applied MathematicsBonn UniversityBonnGermany
  2. 2.Mathematics DepartmentChiba UniversityChibaJapan
  3. 3.Zentrum MathematikTU MünchenGarchingGermany

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