Journal of Statistical Physics

, Volume 125, Issue 4, pp 801–820 | Cite as

Heat Transport in Harmonic Lattices

  • Abhishek DharEmail author
  • Dibyendu Roy


We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green’s function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.


Harmonic crystal quantum Langevin equations non-equilibrium Green’s function Fourier’s law 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. M. Ziman, Principles of the theory of solids, Second Edition (Cambridge University Press, 1972), pp. 71.Google Scholar
  2. 2.
    F. Bonetto, J. L. Lebowitz, and L. Rey-Bellet, Fourier’s law: a challenge to theorists, in A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski (eds.), Mathematical Physics 2000, (Imperial College Press, London, 2000), pp. 128–150.Google Scholar
  3. 3.
    S. Lepri, R. Livi, and A. Politi, Thermal conduction in classical low-dimensional lattices, Phys. Rep. 377:1 (2003).CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    T. Hatano, Heat conduction in the diatomic Toda lattice revisited, Phys. Rev. E 59:R1 (1999); A. Dhar, Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses, Phys. Rev. Lett. 86:3554 (2001); B. Li, H. Zhao, and B. Hu, Can Disorder Induce a Finite Thermal Conductivity in 1D Lattices? Phys. Rev. Lett. 86:63 (2001); P. Grassberger, W. Nadler, and L. Yang, Heat conduction and entropy production in a one-dimensional hard-particle gas, Phys. Rev. Lett. 89:180601 (2002); A. V. Savin, G. P. Tsironis, and A. V. Zolotaryuk, Heat conduction in one-dimensional systems with hard-point interparticle interactions, Phys. Rev. Lett. 88:154301 (2002); G. Casati and T. Prosen, Anomalous heat conduction in a one-dimensional ideal gas, Phys. Rev. E 67:015203(R) (2003); J. S. Wang and B. Li, Intriguing heat conduction of a chain with transverse motions, Phys. Rev. Lett. 92:074302 (2004).CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    S. Lepri, R. Livi, and A. Politi, Heat conduction in chains of nonlinear oscillators, Phys. Rev. Lett. 78:1896 (1997).CrossRefADSGoogle Scholar
  6. 6.
    O. Narayan and S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89:200601 (2002).CrossRefADSGoogle Scholar
  7. 7.
    A. Dhar, Heat conduction in the disordered harmonic chain revisited, Phys. Rev. Lett. 86:5882 (2001).CrossRefADSGoogle Scholar
  8. 8.
    S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1995).Google Scholar
  9. 9.
    Y. Imry, Introduction to Mesoscopic Physics (Oxford University Press, 1997).Google Scholar
  10. 10.
    H. Spohn, The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics, math-ph/0505025.Google Scholar
  11. 11.
    Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a harmonic crystal in a stationary nonequilibrium state, J. Math. Phys. 8:1073 (1967).CrossRefGoogle Scholar
  12. 12.
    R. Rubin and W. Greer, Abnormal lattice thermal conductivity of a one-dimensional, harmonic, isotopically disordered crystal, J. Math. Phys. (N.Y.) 12, 1686 (1971).CrossRefGoogle Scholar
  13. 13.
    A. J. O’Connor and J. L. Lebowitz, Heat conduction and sound transmission in isotopically disordered harmonic crystals, J. Math. Phys. 15:692 (1974).CrossRefMathSciNetGoogle Scholar
  14. 14.
    L. W. Lee and A. Dhar, Heat Conduction in a two-dimensional harmonic crystal with disorder, Phys. Rev. Lett. 95:094302 (2005).CrossRefADSGoogle Scholar
  15. 15.
    G. Y. Hu and R. F. O’Connell, Quantum transport for a many-body system using a quantum Langevin-equation approach, Phys. Rev. B 36:5798 (1987).CrossRefADSGoogle Scholar
  16. 16.
    A. N. Cleland, J. M. Schmidt, and J. Clarke, Influence of the environment on the Coulomb blockade in submicrometer normal-metal tunnel junctions, Phys. Rev. B 45:2950 (1992); G. Y. Hu and R. F. O’Connell, Charge fluctuations and zero-bias resistance in small-capacitance tunnel junctions, Phys. Rev. B 49:16505 (1994), Phys. Rev. B 46:14219 (1992).CrossRefADSGoogle Scholar
  17. 17.
    Y.-C. Chen, J. L. Lebowitz, and C. Liverani, Dissipative quantum dynamics in a boson bath, Phys. Rev. B 40:4664 (1989).CrossRefADSGoogle Scholar
  18. 18.
    U. Zurcher and P. Talkner, Quantum-mechanical harmonic chain attached to heat baths. II. Nonequilibrium properties, Phys. Rev. A 42:3278 (1990).CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    K. Saito, S. Takesue, and S. Miyashita, Energy transport in the integrable system in contact with various types of phonon reservoirs, Phys. Rev. E 61:2397 (2000).CrossRefADSGoogle Scholar
  20. 20.
    A. Dhar and B. S. Shastry, Quantum transport using the Ford-Kac-Mazur formalism, Phys. Rev. B 67:195405 (2003).CrossRefADSGoogle Scholar
  21. 21.
    D. Segal, A. Nitzan and P. Hanggi, Thermal conductance through molecular wires, J. Chem. Phys. 119:6840 (2003).CrossRefADSGoogle Scholar
  22. 22.
    A. Dhar and D. Sen, Nonequilibrium Green’s function formalism and the problem of bound states, Phys. Rev. B 73:085119 (2006).CrossRefADSGoogle Scholar
  23. 23.
    P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic Press, 1995).Google Scholar
  24. 24.
    R. Berkovits and S. Feng, Correlations in coherent multiple scattering, Phys. Reps. 238:135 (1994).CrossRefADSGoogle Scholar
  25. 25.
    L. G. C. Rego and G. Kirczenow, Quantized thermal conductance of dielectric quantum wires, Phys. Rev. Lett. 81:232 (1998).CrossRefADSGoogle Scholar
  26. 26.
    M. P. Blencowe, Quantum energy flow in mesoscopic dielectric structures, Phys. Rev. B 59:4992 (1999).CrossRefADSGoogle Scholar
  27. 27.
    T. Yamamoto and K. Watanabe, Nonequilibrium Green’s function approach to phonon transport in defective carbon nanotubes, Phys. Rev. Lett. 96:255503 (2006).CrossRefADSGoogle Scholar
  28. 28.
    Y. Meir and N. S. Wingreen, Landauer formula for the current through an interacting electron region, Phys. Rev. Lett. 68:2512 (1992).CrossRefADSGoogle Scholar
  29. 29.
    A. Kamenev, in Nanophysics: Coherence and Transport (Lecture notes of the Les Houches Summer School 2004).Google Scholar
  30. 30.
    M. Bolsterli, M. Rich, and W. M. Visscher, Simulation of nonharmonic interactions in a crystal by self-consistent reservoirs, Phys. Rev. A 4:1086 (1970).CrossRefADSGoogle Scholar
  31. 31.
    M. Rich and W. M. Visscher, Disordered harmonic chain with self-consistent reservoirs, Phys. Rev. B 11:2164 (1975).CrossRefADSGoogle Scholar
  32. 32.
    W. M. Visscher and M. Rich, Stationary nonequilibrium properties of a quantum-mechanical lattice with self-consistent reservoirs, Phys. Rev. A 12:675 (1975).CrossRefADSGoogle Scholar
  33. 33.
    F. Bonetto, J. L. Lebowitz, and J. Lukkarinen, Fourier’s law for a harmonic crystal with self-consistent reservoirs, J. Stat. Phys. 116:783 (2004).CrossRefMathSciNetGoogle Scholar
  34. 34.
    K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L. Roukes, Measurement of the quantum of thermal conductance, Nature 404:974 (2000).CrossRefADSGoogle Scholar
  35. 35.
    M. Buttiker, Small normal-metal loop coupled to an electron reservoir, Phys. Rev. B 32:1846 (1985).CrossRefADSGoogle Scholar
  36. 36.
    M. Buttiker, Role of quantum coherence in series resistors, Phys. Rev. B 33:3020 (1986).CrossRefADSGoogle Scholar
  37. 37.
    D. Roy and A. Dhar, Electron transport in one-dimensional wires with self-consistent stochastic reservoirs, In preparation.Google Scholar
  38. 38.
    M. Strass, P. Hanggi, and S. Kohler, Nonadiabatic electron pumping: maximal current with minimal noise, Phys. Rev. Lett. 95:130601 (2005).CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Raman Research InstituteBangaloreIndia

Personalised recommendations