Heat Transport in Harmonic Lattices
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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.
KeywordsHarmonic crystal quantum Langevin equations non-equilibrium Green’s function Fourier’s law
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- 1.J. M. Ziman, Principles of the theory of solids, Second Edition (Cambridge University Press, 1972), pp. 71.Google Scholar
- 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
- 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
- 8.S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1995).Google Scholar
- 9.Y. Imry, Introduction to Mesoscopic Physics (Oxford University Press, 1997).Google Scholar
- 10.H. Spohn, The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics, math-ph/0505025.Google Scholar
- 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
- 23.P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic Press, 1995).Google Scholar
- 29.A. Kamenev, in Nanophysics: Coherence and Transport (Lecture notes of the Les Houches Summer School 2004).Google Scholar
- 37.D. Roy and A. Dhar, Electron transport in one-dimensional wires with self-consistent stochastic reservoirs, In preparation.Google Scholar