Journal of Statistical Physics

, Volume 166, Issue 3–4, pp 841–875 | Cite as

Shocks, Rarefaction Waves, and Current Fluctuations for Anharmonic Chains

  • Christian B. Mendl
  • Herbert Spohn


The nonequilibrium dynamics of anharmonic chains is studied by imposing an initial domain-wall state, in which the two half lattices are prepared in equilibrium with distinct parameters. We analyse the Riemann problem for the corresponding Euler equations and, in specific cases, compare with molecular dynamics. Additionally, the fluctuations of time-integrated currents are investigated. In analogy with the KPZ equation, their typical fluctuations should be of size \(t^{1/3}\) and have a Tracy–Widom GUE distributed amplitude. The proper extension to anharmonic chains is explained and tested through molecular dynamics. Our results are calibrated against the stochastic LeRoux lattice gas.


Fluid dynamics Hydrodynamic waves Statistical mechanics of classical fluids Nonequilibrium thermodynamics 



The work of HS has been supported as a Simons Distinguished Visiting Scholar, when visiting the KITP early 2016. CM acknowledges support from the Alexander von Humboldt Foundation and computing resources of the Leibniz-Rechenzentrum.


  1. 1.
    Rigol, M., Dunjko, V., Olshanii, M.: Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008). doi: 10.1038/nature06838 ADSCrossRefGoogle Scholar
  2. 2.
    Essler, F.H.L., Fagotti, M.: Quench dynamics and relaxation in isolated integrable quantum spin chains (2016). arXiv:1603.06452
  3. 3.
    Bañuls, M.C., Cirac, J.I., Hastings, M.B.: Strong and weak thermalization of infinite nonintegrable quantum systems. Phys. Rev. Lett. 106, 050405 (2011). doi: 10.1103/PhysRevLett.106.050405 ADSCrossRefGoogle Scholar
  4. 4.
    Sabetta, T., Misguich, G.: Nonequilibrium steady states in the quantum XXZ spin chain. Phys. Rev. B 88, 245114 (2013). doi: 10.1103/PhysRevB.88.245114 ADSCrossRefGoogle Scholar
  5. 5.
    Langmann, E., Lebowitz, J.L., Mastropietro, V., Moosavi, P.: Steady states and universal conductance in a quenched Luttinger model. Commun. Math. Phys. (2016). doi: 10.1007/s00220-016-2631-x zbMATHGoogle Scholar
  6. 6.
    Karrasch, C., Ilan, R., Moore, J.E.: Nonequilibrium thermal transport and its relation to linear response. Phys. Rev. B 88, 195129 (2013). doi: 10.1103/PhysRevB.88.195129 ADSCrossRefGoogle Scholar
  7. 7.
    Vasseur, R., Karrasch, C., Moore, J.E.: Expansion potentials for exact far-from-equilibrium spreading of particles and energy. Phys. Rev. Lett. 115, 267201 (2015). doi: 10.1103/PhysRevLett.115.267201 ADSCrossRefGoogle Scholar
  8. 8.
    Castro-Alvaredo, O.A., Doyon, B., Yoshimura, T.: Emergent hydrodynamics in integrable quantum systems out of equilibrium (2016). arXiv:1605.07331
  9. 9.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–1227 (2014). doi: 10.1007/s10955-014-0933-y ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mendl, C.B., Spohn, H.: Dynamic correlators of Fermi-Pasta-Ulam chains and nonlinear fluctuating hydrodynamics. Phys. Rev. Lett. 111, 230601 (2013). doi: 10.1103/PhysRevLett.111.230601 ADSCrossRefGoogle Scholar
  11. 11.
    Lepri, S. (ed.): Thermal Transport in Low Dimensions From Statistical Physics to Nanoscale Heat Transfer. Lecture Notes in Physics 921. Springer, Berlin (2016)Google Scholar
  12. 12.
    Pettini, M., Casetti, L., Cerruti-Sola, M., Franzosi, R., Cohen, E.G.D.: Weak and strong chaos in Fermi-Pasta-Ulam models and beyond. Chaos 15, 015106 (2005). doi: 10.1063/1.1849131 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bressan, A.: Hyperbolic conservation laws: an illustrated tutorial, pp. 157–245. In: Modelling and Optimisation of Flows on Networks, Cetraro, Italy. Lecture Notes in Mathematics 2062 2013. Springer, Berlin (2009). doi: 10.1007/978-3-642-32160-3_2
  14. 14.
    Fritz, J., Tóth, B.: Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas. Commun. Math. Phys. 249, 1–27 (2004). doi: 10.1007/s00220-004-1103-x ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Serre, D.: Systems of Conservation Laws 1: Hyperbolicity, Entropies. Shock Waves. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Serre, D.: Systems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial-Boundary Value Problems. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  17. 17.
    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–L51 (1998). doi: 10.1088/0305-4470/31/2/001 ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Ferrari, P., Sasamoto, T., Spohn, H.: Coupled Kardar–Parisi–Zhang equations in one dimension. J. Stat. Phys. 153, 377–399 (2013). doi: 10.1007/s10955-013-0842-5 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Temple, B.: Global solution of the Cauchy problem for a class of \(2 \times 2\) nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3, 335–375 (1982). doi: 10.1016/S0196-8858(82)80010-9 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mendl, C.B., Spohn, H.: Searching for the Tracy-Widom distribution in nonequilibrium processes. Phys. Rev. E 93, 060101(R) (2016). doi: 10.1103/PhysRevE.93.060101 ADSCrossRefGoogle Scholar
  21. 21.
    Bethe, H.A.: On the theory of shock waves for an arbitrary equation of state, pp. 421–495. In: Classic Papers in Shock Compression Science. Springer, Berlin (1998). doi: 10.1007/978-1-4612-2218-7_11
  22. 22.
    Lax, P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957). doi: 10.1002/cpa.3160100406 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965). doi: 10.1002/cpa.3160180408 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989). doi: 10.1103/RevModPhys.61.75 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fermi, E., Pasta, J., Ulam, S.: Studies of non-linear problems (LA-1940). Technical report, Los Alamos Scientific Laboratory (1955)zbMATHGoogle Scholar
  26. 26.
    Hurtado, P.I.: Breakdown of hydrodynamics in a simple one-dimensional fluid. Phys. Rev. Lett. 96, 010601 (2006). doi: 10.1103/PhysRevLett.96.010601 ADSCrossRefGoogle Scholar
  27. 27.
    Balázs, M., Nagy, A.L., Tóth, B., Tóth, I.: Coexistence of shocks and rarefaction fans: complex phase diagram of a simple hyperbolic particle system (2016). arXiv:1601.02161
  28. 28.
    Mendl, C.B., Spohn, H.: Equilibrium time-correlation functions for one-dimensional hard-point systems. Phys. Rev. E 90, 012147 (2014). doi: 10.1103/PhysRevE.90.012147 ADSCrossRefGoogle Scholar
  29. 29.
    Wendroff, B.: The Riemann problem for materials with nonconvex equations of state I: isentropic flow. J. Math. Anal. Appl. 38, 454–466 (1972). doi: 10.1016/0022-247X(72)90103-5 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000). doi: 10.1007/s002200050027 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Phys. Lett. B 305, 115–118 (1993). doi: 10.1016/0370-2693(93)91114-3 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994). doi: 10.1007/BF02100489 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process, pp. 185–204. In: In and Out of Equilibrium: Probability with a Physics Flavor, Progress in Probability, vol. 51. Birkhäuser Boston (2002). doi: 10.1007/978-1-4612-0063-5_7
  34. 34.
    Ben Arous, G., Corwin, I.: Current fluctuations for TASEP: a proof of the Prähofer–Spohn conjecture. Ann. Probab. 39, 104–138 (2011). doi: 10.1214/10-aop550 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009). doi: 10.1007/s00220-009-0761-0 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mendl, C.B., Spohn, H.: Current fluctuations for anharmonic chains in thermal equilibrium. J. Stat. Mech. 2015, P03007 (2015). doi: 10.1088/1742-5468/2015/03/P03007 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Geballe Laboratory for Advanced MaterialsStanford UniversityStanfordUSA
  2. 2.Stanford Institute for Materials and Energy SciencesSLAC National Accelerator LaboratoryMenlo ParkUSA
  3. 3.Zentrum Mathematik and Physik DepartmentTechnische Universität MünchenGarching bei MünchenGermany

Personalised recommendations