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Heat Conduction in the Frenkel-Kontorova Model

Heat Conduction in the Frenkel-Kontorova Model. “It seems there is no problem in modern physics for which there are on record as many false starts, and as many theories which overlook some essential feature, as in the problem of the thermal conductivity of nonconducting crystals.”

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Heat Conduction in the Frenkel-Kontorova Model

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  1. Heat Conduction in the Frenkel-Kontorova Model

  2. “It seems there is no problem in modern physics for which there are on record as many false starts, and as many theories which overlook some essential feature, as in the problem of the thermal conductivity of nonconducting crystals.” R.E. Peierls

  3. T+ T-

  4. Fourier Law (1808)

  5. Hamiltonian The crux of the heat conduction problem lies in low dimensions, i.e., one and two dimensions.

  6. Green-Kubo Formula Mode coupling theory • d =1 • d =2 • d =3

  7. Some outstanding problems in heat conduction • First-principle derivation of the Fourier law from statistical mechanics. • Necessary and sufficient conditions for the validity of the Fourier law. • Exceptions to the Prosen-Campbell theorem. • Controversy over the exponent and the problem of universality. • Heat conduction in two dimensions. • Practical designs of thermal devices.

  8. Temperature

  9. Heat flux

  10. Density fluctuations

  11. Steady State Total heat flux:

  12. Heat baths 1. Stochastic (Langevin)

  13. 2．Deterministic (Nosé-Hoover)

  14. Temperature gradient can’t be formed. diverges. Fourier law is not obeyed. • Integrable models

  15. B. Non-integrable models 1. No on-site potential, momentum is conserved. Ex. Fermi-Pasta-Ulam (FPU) Diatomic Toda Heisenberg chain

  16. Fourier law is not obeyed.

  17. 2. With on-site potential, momentum is not conserved. Ex. FK [B. Hu, B. Li, H. Zhao (1998)] Ding a Ling

  18. is finite and Fourier law is obeyed.

  19. Necessary and sufficient conditions for normal and anomalous heat conduction A．Normal heat conduction 1. Chaos is neither a necessary nor sufficient condition 2. Necessary conditions (a) On-site potential or vanishing pressure (b) Anharmonicity (c) Non-integrability B．Anomalous heat conduction Momentum conservation is not a necessary but a sufficient condition provided the pressure is non-vanishing.

  20. Controversy over the critical exponent 1. Renormalization group(O. Narayan and S. Ramaswamy): 2. Mode coupling (S. Lepri, R. Livi, and A. Politi):

  21. : spring potential Frenkel – Kontorova (FK) model : external potential

  22. : natural length of spring Standard FK (1938)

  23. Thermal Diodes

  24. I. Relevant factors INT: Integrability DIS: Disorder CS: Chaos OP: On-site potential MC: Momentum conservation PJ: Phase jump TG: Temperature gradient HC: Heat Conduction IP: Inter-particle potential

  25. II. Models HC: Harmonic chain FK: SHG: Sinh-Gorden BSW: Bounded Single-well

  26. III. Remarks • Ref. 2 claimed that 2D disordered harmonic web can have a normal heat condition with sufficiently strong disorder. We have doubts. • Ref. 7 claimed that Diatomic Toda model can have a normal heat conduction. However the latest conclusion in Ref. 8 is opposite. • At low temperature, with the Nose-Hoover heat bath. • Ref. 13 claimed that the hard-point model with alternating masses has a finite heat conduction. Ref. 14 and 15 gave a different conclusion. Ref. 16 studied a modified version of the hard-point model, which can have a finite heat conduction in certain temperature regimes. • FK model can have normal and abnormal heat conductions in different parameters and temperature regimes. See Ref. 19. • Disordered right triangle model and irrational triangle model.

  27. IV. References • S. Lepri, R Livi and A. Politi, Phys. Reports 377, 1 (2003). • L. Yang, Phys. Rev. Lett. 88, 4 (2002). • S. Lepri, Europhys. J. B 18, 411 (2000). • K. Aoki and D. Kusezov, Phys. Rev. Lett. 86, 4029 (2001). • S. Lepri, R Livi and A. Politi, Phys. Rev. Lett. 78, 1896 (1997). • B. Hu, B. W. Li, H. Zhao, Phys. Rev. E 61, 3828 (2000). • E. A. Jackson and A. D. Mistriotis, J. Phys.: CondensMatter 1, 1223 (1989). • T. Hatano, Phys. Rev. E 59, 1 (1999). • B. W. Li, H. Zhao, and B. Hu, Phys. Rev. Lett. 86, 63 (2001). • O. V. Gendelman and A. V. Savin, Phys. Rev. Lett. 84, 2381 (2000). • C. Giardina, R. Livi, A. Piliti and M. Vassalli, Phys. Rev. Lett. 84, 2144 (2000). • A. Dhar, Phys. Rev. Lett. 86, 3554 (2001).

  28. P. L. Garrido, P. I. Hurtado and B. Nadrowski, Phys. Rev. Lett. 86, 5496 (2001). • A. Dhar, Phys. Rev. Lett. 88, 249401 (2002). • G. Casati and T. Prosen,Phys. Rev. E67, 015203 (2003). • A. V. Savin, G. P. Tsironis and A. V. Zolotaryuk, Phys, Rev. Lett. 88, 154301 (2002). • G. Casati, J. Ford, F. Vivaldi and W. M. Visscher, Phys. Rev. Lett. 52, 1861 (1984). • T. Prosen and M. Robnik, J. Phys. A: Math Gen. 25, 3449 (1992). • A. V. Savin and O. V. Gendelman, Phys. Rev. E 67, 041205 (2003). • D. Chen, S. Aubry and G. P. Tsironis, Phys. Rev. Lett. 77, 4776 (1996). • B. Hu, B. W. Li, and H. Zhao, Phys. Rev. E 57, 2992 (1998). • G. P. Tsironis, A. R. Bishop, A. V. Savin and A. V. Zolotaryuk, Phys. Rev. E 60, 6610 (1999). • D. Alonso, R. Artuso, G. Casati and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999). • B. W. Li, L. Wang and B. Hu, Phys. Rev. Lett. 88, 223901 (2002).

  29. Asymmetric Heat Conduction in the Frenkel-Kontorova Model

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