Skip to main content
SpringerLink
Log in
Book cover

Thermal Transport in Low Dimensions pp 215–237Cite as

Thermal Conductivity in Harmonic Lattices with Random Collisions

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Physics ((LNP,volume 921))

Abstract

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    In the non-linear cases we cannot expect that the two step approach would give the same result of the direct rescaling of the dynamics. In the β-FPU the kinetic limit seems to give a different superdiffusion scaling than the direct limit [33, 41].

  2. 2.

    In [15] only the equilibrium fluctuations are considered but the methods developed in [26, 27] can be applied also to the models considered in this section.

References

  1. Ajanki, O., Huveneers, F.: Rigorous scaling law for the heat current in disordered harmonic chain. Commun. Math. Phys. 301, 841–883 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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-AIHP554

    Article  MathSciNet  MATH  Google Scholar 

  3. Basile, G., Bovier, A.: Convergence of a kinetic equation to a fractional diffusion equation. Markov Proc. Rel. Fields 16, 15–44 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Basile, G., Olla, S.: Energy diffusion in harmonic system with conservative noise. J. Stat. Phys. 155(6), 1126–1142 (2014). doi:10.1007/s10955-013-0908-4

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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.96.204303

    Article  Google Scholar 

  6. Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conservative model. Commun. Math. Phys. 287, 67–98 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Basile, G., Olla, S., Spohn, H.: Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. 195(1), 171–203 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Bernardin, C.: Thermal conductivity for a noisy disordered harmonic chain. J. Stat. Phys. 133(3), 417–433 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Bernardin, C., Olla, S.: Thermodynamics and non-equilibrium macroscopic dynamics of chains of anharmonic oscillators. Lecture Notes (2014). Available at https://www.ceremade.dauphine.fr/~olla/

  14. Bernardin, C., Stoltz, G.: Anomalous diffusion for a class of systems with two conserved quantities. Nonlinearity 25(4), 1099–1133 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(8), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Casher, A., Lebowitz, J.L.: Heat flow in regular and disordered harmonic chains. J. Math. Phys.12, 1701–1711 (1971)

    Article  ADS  Google Scholar 

  19. Dhar, A.: Heat transport in low dimensional systems. Adv. Phys. 57(5), 457–537 (2008)

    Article  ADS  Google Scholar 

  20. Dhar, A., Lebowitz, J.L.: Effect of phonon–phonon interactions on localization. Phys. Rev. Lett. 100, 134301 (2008)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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/s00205-014-0741-1

    Article  MathSciNet  MATH  Google Scholar 

  23. Fritz, J., Funaki, T., Lebowitz, J.L.: Stationary states of random Hamiltonian systems. Probab. Theory Relat. Fields 99, 211–236 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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/s00220-015-2417-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Komorowski, T., Olla, S.: Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators. Nonlinearity (2016). arXiv:1506.06465

    Google Scholar 

  28. Komorowski, T., Olla, S.: Diffusive propagation of energy in a non-acoustic chain (2016, preprint)

    Google Scholar 

  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)

    ADS  Google Scholar 

  30. Lepri, S., Livi, R., Politi, A.: Heat conduction in chains of nonlinear oscillators. Phys. Rev. Lett. 78, 1896 (1997)

    Article  ADS  Google Scholar 

  31. Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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.20243

    Article  MathSciNet  MATH  Google Scholar 

  34. Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal. 199(2), 493–525 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Oganesyan, V., Pal, A., Huse, D.: Energy transport in disordered classical spin chains. Phys. Rev. B 80, 115104 (2009)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Peierls, R.E.: Zur kinetischen Theorie der Waermeleitung in Kristallen. Ann. Phys. Lpz. 3, 1055–1101 (1929)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  40. Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124(2–4), 1041–1104 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154(5), 1191–1227 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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/s10955-015-1214-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Commun. Pure Appl. Math. 24, 147–225 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Xu, Y., Li, Z., Duan, W.: Thermal and thermoelectric properties of graphene. Small 10(11), 2182–2199 (2014). doi:10.1002/smll.201303701

    Article  MathSciNet  Google Scholar 

  47. Zola, A., Rosso, A., Kardar, M.: Fractional Laplacian in a bounded interval. Phys. Rev. E 76, 21116 (2007)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy

    Giada Basile

  2. Laboratoire J.A. Dieudonné UMR CNRS 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Cédric Bernardin

  3. IMPA, Rio de Janeiro, Brazil

    Milton Jara

  4. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

    Tomasz Komorowski

  5. Ceremade, UMR CNRS 7534, Université Paris Dauphine, 75775, Paris Cedex 16, France

    Stefano Olla

Authors
  1. Giada Basile
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cédric Bernardin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milton Jara
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tomasz Komorowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Stefano Olla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Olla .

Editor information

Editors and Affiliations

  1. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Sesto Fiorentino (FI), Italy

    Stefano Lepri

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Basile, G., Bernardin, C., Jara, M., Komorowski, T., Olla, S. (2016). Thermal Conductivity in Harmonic Lattices with Random Collisions. In: Lepri, S. (eds) Thermal Transport in Low Dimensions. Lecture Notes in Physics, vol 921. Springer, Cham. https://doi.org/10.1007/978-3-319-29261-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation