Journal of Statistical Physics

, Volume 132, Issue 1, pp 1–33 | Cite as

Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator Chains

  • G. R. Lee-DadswellEmail author
  • B. G. Nickel
  • C. G. Gray


We examine the thermal conductivity and bulk viscosity of a one-dimensional (1D) chain of particles with cubic-plus-quartic interparticle potentials and no on-site potentials. This system is equivalent to the FPU-α β system in a subset of its parameter space. We identify three distinct frequency regimes which we call the hydrodynamic regime, the perturbative regime and the collisionless regime. In the lowest frequency regime (the hydrodynamic regime) heat is transported ballistically by long wavelength sound modes. The model that we use to describe this behaviour predicts that as ω→0 the frequency dependent bulk viscosity, \(\hat{\zeta}(\omega)\) , and the frequency dependent thermal conductivity, \(\tilde{\kappa}(\omega)\) , should diverge with the same power law dependence on ω. Thus, we can define the bulk Prandtl number, \(Pr_{\zeta}=k_{B}\hat{\zeta}(\omega)/(m\hat{\kappa }(\omega))\) , where m is the particle mass and k B is Boltzmann’s constant. This dimensionless ratio should approach a constant value as ω→0. We use mode-coupling theory to predict the ω→0 limit of Pr ζ . Values of Pr ζ obtained from simulations are in agreement with these predictions over a wide range of system parameters. In the middle frequency regime, which we call the perturbative regime, heat is transported by sound modes which are damped by four-phonon processes. This regime is characterized by an intermediate-frequency plateau in the value of \(\hat{\kappa}(\omega)\) . We find that the value of \(\hat{\kappa}(\omega)\) in this plateau region is proportional to T −2 where T is the temperature; this is in agreement with the expected result of a four-phonon Boltzmann-Peierls equation calculation. The Boltzmann-Peierls approach fails, however, to give a nonvanishing bulk viscosity for all FPU-α β chains. We call the highest frequency regime the collisionless regime since at these frequencies the observing times are much shorter than the characteristic relaxation times of phonons.


Transport coefficients 1D systems Classical lattices 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akhieser, A.: On the absorption of sound in solids. J. Phys. (USSR) 1, 277–287 (1939) Google Scholar
  2. 2.
    Alder, B.J., Wainwright, T.E.: Velocity autocorrelations for hard spheres. Phys. Rev. Lett. 18, 988–990 (1967) CrossRefADSGoogle Scholar
  3. 3.
    Alder, B.J., Wainwright, T.E., Gass, D.M.: Decay of time correlations in two dimensions. Phys. Rev. A 4, 233–237 (1971) CrossRefADSGoogle Scholar
  4. 4.
    Ashcroft, N.W., Mermin, N.D.: In: Solid State Physics, p. 507. Harcourt Brace, Orlando (1976) Google Scholar
  5. 5.
    Ashcroft, N.W., Mermin, N.D.: In: Solid State Physics, p. 493. Harcourt Brace, Orlando (1976) Google Scholar
  6. 6.
    Ashcroft, N.W., Mermin, N.D.: In: Solid State Physics, p. 508. Harcourt Brace, Orlando (1976). Problem 3 Google Scholar
  7. 7.
    Casati, G.: Controlling the heat flow: now it is possible. Chaos 15, 015120 (2005) CrossRefADSGoogle Scholar
  8. 8.
    Cipriani, P., Denisov, S., Politi, A.: From anomalous energy diffusion to Levy walks and heat conductivity in one-dimensional systems. Phys. Rev. Lett. 94, 244–301 (2005) CrossRefGoogle Scholar
  9. 9.
    Delfini, L., Lepri, S., Livi, R., Politi, A.: Anomalous kinetics and transport from 1D self-consistent mode-coupling theory. J. Stat. Mech. P02007, 1–17 (2007) Google Scholar
  10. 10.
    Denisov, S., Klafter, J., Urbakh, M.: Dynamical heat channels. Phys. Rev. Lett. 91, 194–301 (2003) CrossRefGoogle Scholar
  11. 11.
    DeVault, G.P.: Lowest nonvanishing contribution to lattice viscosity. Phys. Rev. 155, 875–882 (1967) CrossRefADSGoogle Scholar
  12. 12.
    Ernst, M.H., Hauge, E.H., van Leeuwen, J.M.J.: Asymptotic time behaviour of correlation functions. I. Kinetic terms. Phys. Rev. A 1, 2055–2065 (1971) CrossRefADSGoogle Scholar
  13. 13.
    Ernst, M.H., Hauge, E.H., van Leeuwen, J.M.J.: Asymptotic time behaviour of correlation functions. II. Kinetic and potential terms. J. Stat. Phys. 15, 7–22 (1976) CrossRefADSGoogle Scholar
  14. 14.
    Ernst, M.H., Hauge, E.H., van Leeuwen, J.M.J.: Asymptotic time behaviour of correlation functions. III. Local equilibrium and mode-coupling theory. J. Stat. Phys. 15, 23–58 (1976) CrossRefADSGoogle Scholar
  15. 15.
    Ford, J.: The Fermi-Pasta-Ulam problem: paradox turns discovery. Phys. Rep. 213, 271–310 (1992) CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Forster, D.: Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions. Benjamin, Reading (1975) Google Scholar
  17. 17.
    Grassberger, P., Nagler, W., Yang, L.: Heat conduction and entropy production in a one-dimensional hard-particle gas. Phys. Rev. Lett. 89, 180601 (2002) CrossRefADSGoogle Scholar
  18. 18.
    Gurevich, V.L.: In: Transport in Phonon Systems, p. 244. North Holland, Amsterdam (1986) Google Scholar
  19. 19.
    Lee-Dadswell, G.R.: Prediction of transport coefficients in one-dimensional systems. Ph.D. Thesis, University of Guelph (2005) Google Scholar
  20. 20.
    Lee-Dadswell, G.R., Nickel, B.G., Gray, C.G.: Thermal conductivity and bulk viscosity in quartic oscillator chains. Phys. Rev. E 72, 031202 (2005) CrossRefADSGoogle Scholar
  21. 21.
    Lepri, S., Livi, R., Politi, A.: On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett. 43, 271–276 (1998) CrossRefADSGoogle Scholar
  22. 22.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003) CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Lepri, S., Livi, R., Politi, A.: Universality of anomalous one-dimensional heat conductivity. Phys. Rev. E 68, 067102 (2003) CrossRefADSGoogle Scholar
  24. 24.
    Lifshitz, E.M., Pitaevskii, L.P.: Physical Kinetics. Butterworth-Heinemann, Oxford (1981) Google Scholar
  25. 25.
    Lukkarinen, J., Spohn, H.: Anomalous energy transport in the FPU-beta chain. ArXiv:0704.1607
  26. 26.
    Narayan, O., Ramaswamy, S.: Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89, 200601 (2002) CrossRefADSGoogle Scholar
  27. 27.
    Nickel, B.G.: The solution to the 4-phonon Boltzmann equation for a 1-d chain in a thermal gradient. J. Phys. A 40, 1219–1238 (2007) zbMATHADSMathSciNetGoogle Scholar
  28. 28.
    Omelyan, I.P., Mryglod, I.M., Folk, R.: Symplectic analytically integrable decomposition algorithms: classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations. Comput. Phys. Commun. 151, 272–314 (2003) CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Peierls, R.E.: On the kinetic theory of thermal conduction in crystals. In: Dalitz, R.H., Peierls, R.E. (eds.) Selected Scientific Papers of Rudoph Peierls with Commentary, pp. 15–48. World Scientific, Singapore (1997). Originally published in German in Ann. Phys. 3, 1055–1101 (1929) Google Scholar
  30. 30.
    Peierls, R.E.: Quantum Theory of Solids. Oxford University Press, London (1955) zbMATHGoogle Scholar
  31. 31.
    Pereverzev, A.: Fermi-Pasta-Ulam β lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E 68, 056124 (2003) CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Pomeau, Y., Résibois, P.: Time dependent correlation functions and mode-mode coupling theories. Phys. Rep. 19, 63–139 (1975) CrossRefADSGoogle Scholar
  33. 33.
    Prosen, T., Campbell, D.K.: Momentum conservation implies anomalous energy transport in 1D classical lattices. Phys. Rev. Lett. 84, 2857–2860 (2000) CrossRefADSGoogle Scholar
  34. 34.
    Santhosh, G., Kumar, D.: Anomalous thermal conduction in one dimension: a quantum calculation. Phys. Rev. E 76, 021105 (2007) CrossRefADSGoogle Scholar
  35. 35.
    Toda, M., Kubo, R., Saitô, N.: Statistical Physics I: Equilibrium Statistical Mechanics. Springer, Berlin (1983) Google Scholar
  36. 36.
    Wang, J.S., Li, B.: Intriguing heat conduction of a chain with transverse motions. Phys. Rev. Lett. 92, 074302 (2004) CrossRefADSGoogle Scholar
  37. 37.
    Woodruff, T.O., Ehrenreich, H.: Absorption of sound in insulators. Phys. Rev. 123, 1553–1559 (1961) zbMATHCrossRefADSGoogle Scholar
  38. 38.
    Yoshida, T.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990) CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Yu, C., Shi, L., Yao, Z., Li, D., Majumdar, A.: Thermal conductance and thermopower of an individual single-wall carbon nanotube. Nano Lett. 5, 1842–1846 (2005) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • G. R. Lee-Dadswell
    • 1
    Email author
  • B. G. Nickel
    • 2
  • C. G. Gray
    • 2
  1. 1.Department of Math., Physics and GeologyCape Breton UniversitySydneyCanada
  2. 2.Department of PhysicsUniversity of GuelphGuelphCanada

Personalised recommendations