Advertisement

Journal of Statistical Physics

, Volume 160, Issue 4, pp 861–884 | Cite as

Nonlinear Fluctuating Hydrodynamics in One Dimension: The Case of Two Conserved Fields

  • Herbert Spohn
  • Gabriel Stoltz
Article

Abstract

We study the BS model, which is a one-dimensional lattice field theory taking real values. Its dynamics is governed by coupled differential equations plus random nearest neighbor exchanges. The BS model has two locally conserved fields. The peak structure of their steady state space–time correlations is determined through numerical simulations and compared with nonlinear fluctuating hydrodynamics, which predicts a traveling peak with KPZ scaling function and a standing peak with a scaling function given by the maximally asymmetric Lévy distribution with parameter \(\alpha = 5/3\). As a by-product, we completely classify the universality classes for two coupled stochastic Burgers equations with arbitrary coupling coefficients.

Keywords

KPZ equation Mode-coupling theory Thermal transport in one dimensional systems 

Notes

Acknowledgments

We thank Christian Mendl for numerous instructive discussions, as well as Günther Schütz for stimulating comments on a preliminary version of this manuscript. H.S. is grateful for the support through the Institute for Advanced Study, Princeton, where the first steps in this project were accomplished and thanks David Huse for insisting on a complete classification.

References

  1. 1.
    Ernst, M.H., Hauge, E.H., van Leeuwen, J.M.J.: Asymptotic time behavior of correlation functions. II. Kinetic and potential terms. J. Stat. Phys. 15, 7–22 (1976)ADSCrossRefGoogle Scholar
  2. 2.
    Forster, D., Nelson, D.R., Stephen, M.J.: Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16, 732–749 (1977)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Lepri, S., Livi, R., Politi, A.: Heat conduction in chains of nonlinear oscillators. Phys. Rev. Lett. 78, 1896–1899 (1997)ADSCrossRefGoogle Scholar
  4. 4.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57, 457–537 (2008)ADSCrossRefGoogle Scholar
  6. 6.
    van Beijeren, H.: Exact results for anomalous transport in one-dimensional Hamiltonian systems. Phys. Rev. Lett. 108, 180601 (2012)CrossRefGoogle Scholar
  7. 7.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–1227 (2014)MathSciNetADSCrossRefMATHGoogle Scholar
  8. 8.
    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)ADSCrossRefMATHGoogle Scholar
  9. 9.
    Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1, 113001 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borodin, A., Gorin, V.: Lectures on integrable probability. arXiv:1212.3351 (2012)
  11. 11.
    Borodin, A., Petrov, L.: Integrable probability: from representation theory to Macdonald processes. arXiv:1310.8007 (2013)
  12. 12.
    Quastel, J., Remenik, D.: Airy processes and variational problems. arXiv:1301.0750 (2013)
  13. 13.
    Bernardin, C., Stoltz, G.: Anomalous diffusion for a class of systems with two conserved quantities. Nonlinearity 25, 1099–1133 (2012)MathSciNetADSCrossRefMATHGoogle Scholar
  14. 14.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178, 559–664 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Funaki, T., Quastel, J.: KPZ equation, its renormalization and invariant measures. arXiv:1407.7310 (2014)
  16. 16.
    Borodin, A., Corwin, I., Ferrari, P., Vetö, B.: Height fluctuations for the stationary KPZ equation. arXiv:1407.6977 (2014)
  17. 17.
    Imamura, T., Sasamoto, T.: Stationary correlations for the 1D KPZ equation. J. Stat. Phys. 150, 908–939 (2013)MathSciNetADSCrossRefMATHGoogle Scholar
  18. 18.
    Prähofer, M.: Exact scaling functions for one-dimensional stationary KPZ growth. http://www-m5.ma.tum.de/KPZ
  19. 19.
    Prähofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255–279 (2004)ADSCrossRefMATHGoogle Scholar
  20. 20.
    Ferrari, P., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Commun. Math. Phys. 265, 1–44 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  21. 21.
    Ertaş, D., Kardar, M.: Dynamic relaxation of drifting polymers: a phenomenological approach. Phys. Rev. E 48, 1228–1245 (1993)ADSCrossRefGoogle Scholar
  22. 22.
    Mendl, Ch.B., Spohn, H.: Dynamic correlators of Fermi-Pasta-Ulam chains and nonlinear fluctuating hydrodynamics. Phys. Rev. Lett. 111, 230601 (2013)Google Scholar
  23. 23.
    Bernardin, C., Gonçalves, P., Jara, M.: \(3/4\)-superdiffusion in a system of harmonic oscillators perturbed by a conservative noise. arXiv:1402.1562 (2014)
  24. 24.
    Ferrari, P., Sasamoto, T., Spohn, H.: Coupled Kardar-Parisi-Zhang equations in one dimension. J. Stat. Phys. 153, 377–399 (2013)MathSciNetADSCrossRefMATHGoogle Scholar
  25. 25.
    Popkov, V., Schmidt, J., Schütz, G.M.: Superdiffusive modes in two-species driven diffusive systems. Phys. Rev. Lett. 112, 200602 (2014)ADSCrossRefGoogle Scholar
  26. 26.
    Popkov, V., Schmidt, J., Schütz, G.M.: Universality classes in two-component driven diffusive systems. arXiv:1410.8026 (2014)
  27. 27.
    Kulkarni, M., Lamacraft, A.: Finite-temperature dynamical structure factor of the one-dimensional Bose gas: from the Gross-Pitaevskii equation to the Kardar-Parisi-Zhang universality class of dynamical critical phenomena. Phys. Rev. A 88, 021603(R) (2013)ADSCrossRefGoogle Scholar
  28. 28.
    Kulkarni, M., Spohn H., Huse, D.: Nonlinear fluctuating hydrodynamics for the 1D Bose gas, draftGoogle Scholar
  29. 29.
    Mendl, Ch.B., Spohn, H.: Nonlinear lattice Schrödinger equation at low temperatures (in press)Google Scholar
  30. 30.
    Kac, M., van Moerbeke, P.: On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices. Adv. Math. 16, 160–169 (1975)CrossRefMATHGoogle Scholar
  31. 31.
    Toda, M.: Theory of Nonlinear Lattices (second enlarged edition). Solid-State Sciences, vol. 20. Springer, Berlin (1988)Google Scholar
  32. 32.
    Mendl, Ch.B., Spohn, H.: Equilibrium time-correlation functions for one-dimensional hard-point systems. Phys. Rev. E 90, 012147 (2014)Google Scholar
  33. 33.
    Das, S.G., Dhar, A., Saito, K., Mendl, Ch.B., Spohn, H.: Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain. Phys. Rev. E 90, 012124 (2014)Google Scholar
  34. 34.
    Straka, M.: KPZ scaling in the one-dimensional FPU-model. Master Thesis, University of Florence, Italy (2013)Google Scholar
  35. 35.
    Zwillinger, D.: CRC Standard Mathematical Tables and Formulae, vol. 31. CRC Press, Boca Raton (2003)MATHGoogle Scholar
  36. 36.
    Jara, M., Komorowski, T., Olla, S.: Superdiffusion of energy in a chain of harmonic oscillators with noise. arXiv:1402.2988 (2014)
  37. 37.
    Uchaikin, V., Zolotarev, V.: Chance and Stability. Stable Distributions and Applications, Modern Probability and Statistics Series. De Gruyter, Utrecht (1999)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Zentrum Mathematik and Physik DepartmentTU MünchenGarchingGermany
  2. 2.Université Paris-Est, CERMICS (ENPC), INRIAMarne-la-ValléeFrance

Personalised recommendations