Thermal Conductivity in Harmonic Lattices with Random Collisions

  • Giada Basile
  • Cédric Bernardin
  • Milton Jara
  • Tomasz Komorowski
  • Stefano OllaEmail author
Part of the Lecture Notes in Physics book series (LNP, volume 921)


We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.


Wigner Function Wigner Distribution Random Exchange Linear Wave Equation Equilibrium Fluctuation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Herbert Spohn for many inspiring discussions on this subject.

The research of Cédric Bernardin was supported in part by the French Ministry of Education through the grant ANR-EDNHS. The work of Stefano Olla has been partially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953) and by a CNPq grant Sciences Without Frontiers. Tomasz Komorowski acknowledges the support of the Polish National Science Center grant UMO-2012/07/B/ST1/03320.


  1. 1.
    Ajanki, O., Huveneers, F.: Rigorous scaling law for the heat current in disordered harmonic chain. Commun. Math. Phys. 301, 841–883 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basile, G.: From a kinetic equation to a diffusion under an anomalous scaling. Ann. Inst. Henri Poincare Prob. Stat. 50(4), 1301–1322 (2014). doi:10.1214/13-AIHP554MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Basile, G., Bovier, A.: Convergence of a kinetic equation to a fractional diffusion equation. Markov Proc. Rel. Fields 16, 15–44 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Basile, G., Olla, S.: Energy diffusion in harmonic system with conservative noise. J. Stat. Phys. 155(6), 1126–1142 (2014). doi:10.1007/s10955-013-0908-4ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Basile, G., Bernardin, C., Olla, S.: A momentum conserving model with anomalous thermal conductivity in low dimension. Phys. Rev. Lett. 96, 204–303 (2006). doi:10.1103/PhysRevLett.96.204303CrossRefGoogle Scholar
  6. 6.
    Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conservative model. Commun. Math. Phys. 287, 67–98 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Basile, G., Olla, S., Spohn, H.: Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. 195(1), 171–203 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Basko, D.M.: Weak chaos in the disordered nonlinear Schrödinger chain: destruction of Anderson localization by Arnold diffusion. Ann. Phys. 326, 1577–1655 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bernardin, C.: Thermal conductivity for a noisy disordered harmonic chain. J. Stat. Phys. 133(3), 417–433 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bernardin, C., Huveneers, F.: Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential. Probab. Theory Relat. Fields 157(1–2), 301–331 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bernardin, C., Olla, S.: Fourier law and fluctuations for a microscopic model of heat conduction. J. Stat. Phys. 118(3/4), 271–289 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bernardin, C., Olla, S.: Transport properties of a chain of anharmonic oscillators with random flip of velocities. J. Stat. Phys. 145, 1224–1255 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bernardin, C., Olla, S.: Thermodynamics and non-equilibrium macroscopic dynamics of chains of anharmonic oscillators. Lecture Notes (2014). Available at
  14. 14.
    Bernardin, C., Stoltz, G.: Anomalous diffusion for a class of systems with two conserved quantities. Nonlinearity 25(4), 1099–1133 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bernardin, C., Goncalves, P., Jara, M.: 3/4 fractional superdiffusion of energy in a system of harmonic oscillators perturbed by a conservative noise. Arch. Ration. Mech. Anal. 1–38 (2015). doi:10.1007/s00205-015-0936-0. Issn 1432-0673 [online first]Google Scholar
  16. 16.
    Bernardin, C., Goncalves, P., Jara, M., Sasada, M., Simon, M.: From normal diffusion to superdiffusion of energy in the evanescent flip noise limit. J. Stat. Phys. 159(6), 1327–1368 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(8), 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Casher, A., Lebowitz, J.L.: Heat flow in regular and disordered harmonic chains. J. Math. Phys.12, 1701–1711 (1971)ADSCrossRefGoogle Scholar
  19. 19.
    Dhar, A.: Heat transport in low dimensional systems. Adv. Phys. 57(5), 457–537 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    Dhar, A., Lebowitz, J.L.: Effect of phonon–phonon interactions on localization. Phys. Rev. Lett. 100, 134301 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    Dhar, A., Venkateshan, K., Lebowitz, J.L.: Heat conduction in disordered harmonic lattices with energy-conserving noise. Phys. Rev. E 83(2), 021108 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    Even, N., Olla, S.: Hydrodynamic limit for an Hamiltonian system with boundary conditions and conservative noise. Arch. Ration. Mech. Appl. 213, 561–585 (2014). doi:10.1007/s00205-014-0741-1MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fritz, J., Funaki, T., Lebowitz, J.L.: Stationary states of random Hamiltonian systems. Probab. Theory Relat. Fields 99, 211–236 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Huveneers, F.: Drastic fall-off of the thermal conductivity for disordered lattices in the limit of weak anharmonic interactions. Nonlinearity 26(3), 837–854 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jara, M., Komorowski, T., Olla, S.: A limit theorem for an additive functionals of Markov chains. Ann. Appl. Probab. 19(6), 2270–2300 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jara, M., Komorowski, T., Olla, S.: Superdiffusion of energy in a chain of harmonic oscillators with noise. Commun. Math. Phys. 339, 407–453 (2015). doi:10.1007/s00220-015-2417-6ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Komorowski, T., Olla, S.: Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators. Nonlinearity (2016). arXiv:1506.06465Google Scholar
  28. 28.
    Komorowski, T., Olla, S.: Diffusive propagation of energy in a non-acoustic chain (2016, preprint)Google Scholar
  29. 29.
    Kundu, A., Chaudhuri, A., Roy, D., Dhar, A., Lebowitz, J.L., Spohn, H.: Heat transport and phonon localization in mass-disordered harmonic crystals. Phys. Rev. B 81, 064301 (2010)ADSGoogle Scholar
  30. 30.
    Lepri, S., Livi, R., Politi, A.: Heat conduction in chains of nonlinear oscillators. Phys. Rev. Lett. 78, 1896 (1997)ADSCrossRefGoogle Scholar
  31. 31.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Lepri, S., Meija-Monasterio, C., Politi, A.: A stochastic model of anomalous heat transport: analytical solution of the steady state. J. Phys. A: Math. Gen. 42, 025001 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lukkarinen, J., Spohn, H.: Anomalous energy transport in the FPU-β chain. Commun. Pure Appl. Math. 61, 1753–1786 (2008). doi:10.1002/cpa.20243MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal. 199(2), 493–525 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Oganesyan, V., Pal, A., Huse, D.: Energy transport in disordered classical spin chains. Phys. Rev. B 80, 115104 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    Olla, S., Varadhan, S.R.S., Yau, H.T.: Hydrodynamic limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Peierls, R.E.: Zur kinetischen Theorie der Waermeleitung in Kristallen. Ann. Phys. Lpz. 3, 1055–1101 (1929)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of harmonic crystal in a stationary non-equilibrium state. J. Math. Phys. 8, 1073–1078 (1967)ADSCrossRefGoogle Scholar
  39. 39.
    Rubin, R.J., Greer, W.L.: Abnormal lattice thermal conductivity of a one-dimensional, harmonic, isotopically disordered crystal. J. Math. Phys. 12, 1686–1701 (1971)ADSCrossRefGoogle Scholar
  40. 40.
    Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124(2–4), 1041–1104 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154(5), 1191–1227 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Spohn, H.: Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains. In: Lepri, S. (ed.) Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer. Springer, Heidelberg (2016)Google Scholar
  43. 43.
    Spohn, H., Stoltz, G.: Nonlinear fluctuating hydrodynamics in one dimension: the case of two conserved fields. J. Stat. Phys. 160, 861–884 (2015). doi:10.1007/s10955-015-1214-0ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Commun. Pure Appl. Math. 24, 147–225 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Verheggen, T.: Transmission coefficient and heat conduction of a harmonic chain with random masses: asymptotic estimates on products of random matrices. Commun. Math. Phys. 68, 69–82 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Xu, Y., Li, Z., Duan, W.: Thermal and thermoelectric properties of graphene. Small 10(11), 2182–2199 (2014). doi:10.1002/smll.201303701MathSciNetCrossRefGoogle Scholar
  47. 47.
    Zola, A., Rosso, A., Kardar, M.: Fractional Laplacian in a bounded interval. Phys. Rev. E 76, 21116 (2007)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Giada Basile
    • 1
  • Cédric Bernardin
    • 2
  • Milton Jara
    • 3
  • Tomasz Komorowski
    • 4
  • Stefano Olla
    • 5
    Email author
  1. 1.Dipartimento di MatematicaUniversità di Roma La SapienzaRomaItaly
  2. 2.Laboratoire J.A. Dieudonné UMR CNRS 7351Université de Nice Sophia-AntipolisNice Cedex 02France
  3. 3.IMPARio de JaneiroBrazil
  4. 4.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  5. 5.Ceremade, UMR CNRS 7534Université Paris DauphineParis Cedex 16France

Personalised recommendations