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Energy Carriers in the Fermi-Pasta- Ulam β Lattice: Solitons or Phonons ?

Energy Carriers in the Fermi-Pasta- Ulam β Lattice: Solitons or Phonons ?. Nianbei Li Max-Planck-Institut für Physik komplexer Systeme. Collaborate with Prof. Baowen Li and Dr. Sergej Flach. Outline. Introduction Numerical Method Results and Discussions Conclusion. Introduction.

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Energy Carriers in the Fermi-Pasta- Ulam β Lattice: Solitons or Phonons ?

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  1. Energy Carriers in the Fermi-Pasta-Ulamβ Lattice: Solitons or Phonons? Nianbei Li Max-Planck-Institut für Physik komplexer Systeme Collaborate with Prof. Baowen Li and Dr. SergejFlach

  2. Outline • Introduction • Numerical Method • Results and Discussions • Conclusion

  3. Introduction • Linear system: Harmonic Lattice • The only excitation mode and Energy Carriers: Phonons Phonon Spectrum: Sound Velocity:

  4. Introduction • Nonlinear system: FPU-β Lattice • Excitation modes: possible Energy Carriers • Solitons • Effective Phonons

  5. Introduction • Solitons a self-reinforcing solitary wave that maintains its shape while it travels at constant speed from Internet

  6. Introduction • Solitons in FPU-β Lattice where Soliton Velocity: Excited from zero temperature background F. Zhang et. al., PRE 61, 3541 (2000)

  7. Introduction • Solitons in FPU-β Lattice Snapshot at t=300 Soliton Velocity: Excited from zero temperature background H. Zhao, PRL 96, 140602 (2006)

  8. Introduction • Solitons in FPU-β Lattice at finite temperature? • Measure the speed of the propagating front of an energy pulse at specific temperature Solid line: with η= 2 K. Aoki & D. Kusnezov, PRL 86, 4029 (2001)

  9. Introduction • Experimental verification? Thermal Diode built with Nanotubes The authors attribute this thermal rectification effect to the soliton mismatch C. W. Chang et.al., Science 314, 1121 (2006)

  10. Introduction • Effective Phonons C. Alabiso, M. Casartelli and P. Marenzoni, J. Stat. Phys., 79, 451 (1995) C. Alabiso and M. Casartelli, J. Phys. A, 34, 1223 (2001)

  11. Introduction • Spectrum of Effective Phonons Temperature-dependent renormalization

  12. Introduction • Sound Velocity of Effective Phonons Solitons K. Aoki & D. Kusnezov, PRL 86, 4029 (2001) N. Li, P. Tong and B. Li, EPL 75, 49 (2006)

  13. Introduction Solitons Effective Phonons K. Aoki & D. Kusnezov, PRL 86, 4029 (2001) N. Li, P. Tong and B. Li, EPL 75, 49 (2006) A more accurate numerical method is needed

  14. Numerical Method • Measure the Sound Velocity from an equilibrium approach • Evolving of the correlation function of the energy and momentum fluctuations • Energy and momentum conservation H. Zhao, PRL 96, 140602 (2006)

  15. Numerical Method • Spatial distribution of the correlators H. Zhao, PRL 96, 140602 (2006) T=0.5, t=100 (blue), t=200 (green), t=300 (red)

  16. Numerical Method • Sound velocity: speed of the propagating front H. Zhao, PRL 96, 140602 (2006) The slope determines the sound velocity

  17. Results • Snapshot of the Correlators at t=60: • low (T=0.02), intermediate (T=0.5) and high (T=5) temperature regime

  18. Results • Sound velocity: speed of the propagating front Solitons Effective Phonons

  19. Results • Spatiotemporal evolution of energy density relative displacements Harmonic Lattice T=1 FPU-β Lattice T=1 T=20

  20. Results • Hnmodels • H4: high temperature limit of FPU-β lattice • Sound Velocities for Effective Phonons

  21. Results • Sound Velocities forHnmodels

  22. Results • Relation between anomalous heat conduction and superdiffusion in FPU-β lattice • Heat conductivity • Energy diffusion Billiard gas model B. Li and J. Wang, PRL 91, 044301 (2003) Dynamical heat channel S. Denisov, J. Klafter, and M. Urbakh, PRL 91, 194301 (2003)

  23. Results H. Zhao, PRL 96, 140602 (2006) • Probability Distribution Function of energy diffusion Assume zero temperature background: energy density

  24. Results B. Li, J. Wang, L. Wang and G. Zhang Chaos 15, 015121 (2005) H. Zhao, PRL 96, 140602 (2006)

  25. Results Assume the second moment is dominated by the satellite peak area Approximate the area as a delta function with diminishing amplitude:

  26. Results • Green-Kubo formula for the heat conductivity • Harmonic lattice • Approximation: only correlation for long-wave length phonons (k=0) are left

  27. Results • FPU-β lattice • Superdiffusion • Anomalous heat conduction • Relation

  28. Conclusions • Energy carriers in FPU-β lattice are the Effective Phonons instead of the previously believed Solitons. • The relation between Energy diffusion and Anomalous heat conduction inFPU-β lattice follows the prediction from dynamical heat channels. http://arxiv.org/abs/1003.6113 Accepted by PRL

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