Journal of Statistical Physics

, Volume 160, Issue 3, pp 583–621 | Cite as

Anomalous Energy Transport in FPU-\(\beta \) Chain

  • Antoine Mellet
  • Sara Merino-Aceituno


This paper is devoted to the derivation of a macroscopic fractional diffusion equation describing heat transport in an anharmonic chain. More precisely, we study here the so-called FPU-\(\beta \) chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbors by a harmonic potential, weakly perturbed by a quartic potential. The starting point of our mathematical analysis is a kinetic equation: Lattice vibrations, responsible for heat transport, are modeled by an interacting gas of phonons whose evolution is described by the Boltzmann phonon equation. Our main result is the rigorous derivation of an anomalous diffusion equation starting from the linearized Boltzmann phonon equation.


FPU-\(\beta \) chain Boltzmann phonon equation Hydrodynamic limit  Anomalous diffusion 



This material is based upon work supported by the Kinetic Research Network (KI-Net) under the NSF Grant No. RNMS #1107444. A.M. is partially supported by NSF Grant DMS-1201426. S.M thanks the University of Maryland and CSCAMM (Center for Scientific Computation and Mathematical Modeling) for their hospitality; the Cambridge Philosophical Society and Lucy Cavendish College (University of Cambridge) for their financial support. Thanks to Clément Mouhot from the University of Cambridge for his help and support. Thanks to Herbert Spohn from Technische Universität München for useful discussions. S.M is supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.


  1. 1.
    Abdallah, N.B., Mellet, A., Puel, M.: Fractional diffusion limit for collisional kinetic equations: a hilbert expansion approach. Kinet. Relat. Models 4, 873–900 (2011)Google Scholar
  2. 2.
    Abdallah, N.B., Mellet, A., Puel, M.: Anomalous diffusion limit for kinetic equations with degenerate collision frequency. Math. Models Methods Appl. Sci. 21(11), 2249–2262 (2011)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Allaire, G., Golse, F.: Transport et diffusion. Lecture notes, Ecole polytechnique (in French) (2013)Google Scholar
  4. 4.
    Bardos, C., Golse, F., Levermore, D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63(1–2), 323–344 (1991)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Basile, G., Bernardin, C., Olla, S.: Momentum conserving model with anomalous thermal conductivity in low dimensional systems. Phys. Rev. Lett. 96(20), 204303 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conservative model. Commun. Math. Phys. 287(1), 67–98 (2009)MATHMathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Basile, G., Olla, S., Spohn, H.: Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. Anal. 195(1), 171–203 (2010)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Berman, G.P., Izrailev, F.M.: The fermi-pasta-ulam problem: fifty years of progress. Chaos: an interdisciplinary. J. Nonlinear Sci. 15(1), 015104 (2005)MathSciNetGoogle Scholar
  9. 9.
    Bernardin, C., Gonçalves, P., Jara, M.: 3/4 Fractional superdiffusion of energy in a system of harmonic oscillators perturbed by a conservative noise. ArXiv e-prints (2014)Google Scholar
  10. 10.
    Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: A challenge for theorists. arXiv preprint math-ph/0002052 (2000)Google Scholar
  11. 11.
    Bricmont, J., Kupiainen, A.: Approach to equilibrium for the phonon Boltzmann equation. Commun. Math. Phys. 281(1), 179–202 (2008)MATHMathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Debye, P.: Vorträge über die kinetische theorie der wärme. Teubner (1914)Google Scholar
  13. 13.
    Delfini, L., Lepri, S., Livi, R., Politi, A.: Anomalous kinetics and transport from 1d self-consistent mode-coupling theory. J. Stat. Mech. Theory Exp. 2, P02007 (2007)Google Scholar
  14. 14.
    Gerschenfeld, A., Derrida, B., Lebowitz, J.L.: Anomalous Fourier’s law and long range correlations in a 1D non-momentum conserving mechanical model. J. Stat. Phys. 141(5), 757–766 (2010)MATHMathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Hittmeir, S., Merino-Aceituno, S: Kinetic derivation of fractional stokes and stokes-fourier systems. arXiv:1408.6400 (2014)
  16. 16.
    Jara, M., Komorowski, T., Olla, S.: Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19(6), 2270–2300 (2009)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Jara, M., Komorowski, T., Olla, S.: Superdiffusion of energy in a chain of harmonic oscillators with noise. ArXiv e-prints (2014)Google Scholar
  18. 18.
    Lepri, S., Livi, R., Politi, A.: Heat conduction in chains of nonlinear oscillators. Phys. Rev. Lett. 78, 1896 (1997)ADSCrossRefGoogle Scholar
  19. 19.
    Lepri, S., Livi, R., Politi, A.: On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett. 43, 271 (1998)ADSCrossRefGoogle Scholar
  20. 20.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377(1), 1–80 (2003)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Lepri, S., Livi, R., Politi, A.: Universality of anomalous one-dimensional heat conductivity. Phys. Rev. 68, 067102 (2003)Google Scholar
  22. 22.
    Lepri, S., Livi, R., Politi, A.: Studies of thermal conductivity in fermipastaulam-like lattices. Chaos 15, 015118 (2005)ADSCrossRefGoogle Scholar
  23. 23.
    Lepri, S., Livi, R., Politi, A.: Studies of thermal conductivity in fermi-pasta-ulam-like lattices. Chaos: an interdisciplinary. J. Nonlinear Sci. 15(1), 015118 (2005)Google Scholar
  24. 24.
    Lukkarinen, J., Spohn, H.: Anomalous energy transport in the fpu-\(\beta \) chain. Commun. Pure Appl. Math. 61(12), 1753–1786 (2008)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Mellet, A.: Fractional diffusion limit for collisional kinetic equations: a moments method. Indiana Univ. Math. J. 59(4), 1333–1360 (2010)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal. 199(2), 493–525 (2011)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Mouhot, C.: Lecture Notes: Mathematical Topics in Kinetic Theory. Chap. 4: The linear Boltzmann equation (2013)Google Scholar
  28. 28.
    Narayan, O., Ramaswamy, S.: Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89 (2002)Google Scholar
  29. 29.
    Olla, S.: Energy diffusion and superdiffusion in oscillators lattice networks. New Trends in Mathematical Physics, pp. 539–547. Springer, New York (2009)CrossRefGoogle Scholar
  30. 30.
    Peierls, R.: Zur kinetischen theorie der wärmeleitung in kristallen. Annalen der Physik 395(8), 1055–1101 (1929)ADSCrossRefGoogle Scholar
  31. 31.
    Pereverzev, A.: Fermi-Pasta-Ulam \(\beta \) lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E 68(5), 056124 (2003)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Shimada, T., Murakami, T., Yukawa, S., Saito, K., Ito, N.: Simulational study on dimensionality dependence of heat conduction. J. Phys. Soc. Jpn. 69(10), 3150–3153 (2000)ADSCrossRefGoogle Scholar
  33. 33.
    Spohn, H.: Collisional invariants for the phonon boltzmann equation. J. Stat. Phys. 124(5), 1131–1135 (2006)MATHMathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124(2–4), 1041–1104 (2006)MATHMathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Spohn, H.: On the Boltzmann equation for weakly nonlinear wave equations. Boltzmann’s Legacy. ESI Lectures in Mathematics and Physics, pp. 145–159. European Mathematical Society, Zürich (2008)CrossRefGoogle Scholar
  36. 36.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154(5), 1191–1227 (2014)MATHMathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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