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Thermal Transport in Low Dimensions pp 239–274Cite as

Simulation of Heat Transport in Low-Dimensional Oscillator Lattices

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Part of the book series: Lecture Notes in Physics ((LNP,volume 921))

Abstract

The study of heat transport in low-dimensional oscillator lattices presents a formidable challenge. Theoretical efforts have been made trying to reveal the underlying mechanism of diversified heat transport behaviors. In lack of a unified rigorous treatment, approximate theories often may embody controversial predictions. It is therefore of ultimate importance that one can rely on numerical simulations in the investigation of heat transfer processes in low-dimensional lattices. The simulation of heat transport using the non-equilibrium heat bath method and the Green-Kubo method will be introduced. It is found that one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) momentum-conserving nonlinear lattices display power-law divergent, logarithmic divergent and constant thermal conductivities, respectively. Next, a novel diffusion method is also introduced. The heat diffusion theory connects the energy diffusion and heat conduction in a straightforward manner. This enables one to use the diffusion method to investigate the objective of heat transport. In addition, it contains fundamental information about the heat transport process which cannot readily be gathered otherwise.

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References

  1. James, M.L., Smith, G.M., Wolford, J.C.: Applied Numerical Methods for Digital Computation. HarperCollins College Publishers, New York (1993)

    MATH  Google Scholar 

  2. Dormand, J.R., El-Mikkawy, M.E.A., Prince, P.J.: IMA J. Numer. Anal. 7, 423 (1987)

    Article  MathSciNet  Google Scholar 

  3. Dormand, J.R., El-Mikkawy, M.E.A., Prince, P.J.: IMA J. Numer. Anal. 11, 297 (1991)

    Article  MathSciNet  Google Scholar 

  4. Lepri, S., Livi, R., Politi, A.: Phys. Rep. 377, 1 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  5. Dhar, A.: Adv. Phys. 57(5), 457 (2008)

    Article  ADS  Google Scholar 

  6. Narayan, O., Ramaswamy, S.: Phys. Rev. Lett. 89, 200601 (2002)

    Article  ADS  Google Scholar 

  7. Mai, T., Narayan, O.: Phys. Rev. E 73, 061202 (2006)

    Article  ADS  Google Scholar 

  8. Wang, J.S., Li, B.: Phys. Rev. Lett. 92, 074302 (2004)

    Article  ADS  Google Scholar 

  9. Wang, J.S., Li, B.: Phys. Rev. E 70, 021204 (2004)

    Article  ADS  Google Scholar 

  10. Delfini, L., Lepri, S., Livi, R., Politi, A.: Phys. Rev. E 73, 060201 (2006)

    Article  ADS  Google Scholar 

  11. Delfini, L., Lepri, S., Livi, R., Politi, A.: J. Stat. Mech. Theory Exp. 2007(02), P02007 (2007)

    Article  Google Scholar 

  12. Delfini, L., Denisov, S., Lepri, S., Livi, R., Mohanty, P.K., Politi, A.: Eur. Phys. J. Special Topics 146, 21 (2007)

    Article  ADS  Google Scholar 

  13. Santhosh, G., Kumar, D.: Phys. Rev. E 77, 011113 (2008)

    Article  ADS  Google Scholar 

  14. Liu, S., Liu, J., Hänggi, P., Wu, C., Li, B.: Phys. Rev. B 90, 174304 (2014)

    Article  ADS  Google Scholar 

  15. Pereverzev, A.: Phys. Rev. E 68, 056124 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  16. Lukkarinen, J., Spohn, H.: Commun. Pure Appl. Math. 61(12), 1753 (2008)

    Article  MathSciNet  Google Scholar 

  17. Lepri, S., Livi, R., Politi, A.: Phys. Rev. Lett. 78(10), 1896 (1997)

    Article  ADS  Google Scholar 

  18. Mai, T., Dhar, A., Narayan, O.: Phys. Rev. Lett. 98(18), 184301 (2007)

    Article  ADS  Google Scholar 

  19. Delfini, L., Lepri, S., Livi, R., Politi, A.: Phys. Rev. Lett. 100, 199401 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  20. Dhar, A., Narayan, O.: Phys. Rev. Lett. 100, 199402 (2008)

    Article  ADS  Google Scholar 

  21. Prosen, T., Campbell, D.K.: Phys. Rev. Lett. 84, 2857 (2000)

    Article  ADS  Google Scholar 

  22. Gendelman, O.V., Savin, A.V.: Phys. Rev. Lett. 84(11), 2381 (2000)

    Article  ADS  Google Scholar 

  23. Wang, L., Wang, T.: Europhys. Lett. 93, 54002 (2011)

    Article  ADS  Google Scholar 

  24. Delfini, L., Lepri, S., Livi, R., Mejia-Monasterio, C., Politi, A.: J. Phys. A Math. Theor. 43(14), 145001 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  25. Zhong, Y., Zhang, Y., Wang, J., Zhao, H.: Phys. Rev. E 85, 060102 (2012)

    Article  ADS  Google Scholar 

  26. Wang, L., Hu, B., Li, B.: Phys. Rev. E 88, 052112 (2013)

    Article  ADS  Google Scholar 

  27. Savin, A.V., Kosevich, Y.A.: Phys. Rev. E 89, 032102 (2014)

    Article  ADS  Google Scholar 

  28. Das, S.G., Dhar, A., Narayan, O.: J. Stat. Phys 154, 204 (2014)

    Article  MathSciNet  Google Scholar 

  29. Chen, S., Wang, J., Casati, G., Benenti, G.: Phys. Rev. E 90, 032134 (2014)

    Article  ADS  Google Scholar 

  30. Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II. Springer Series in Solid State Sciences, vol. 31. Springer, Berlin (1991)

    Google Scholar 

  31. Chen, S., Zhang, Y., Wang, J., Zhao, H.: Phys. Rev. E 89, 022111 (2014)

    Article  ADS  Google Scholar 

  32. Wang, L., Hu, B., Li, B.: Phys. Rev. E 86, 040101 (2012)

    Article  ADS  Google Scholar 

  33. Wang, L., He, D., Hu, B.: Phys. Rev. Lett. 105, 160601 (2010)

    Article  ADS  Google Scholar 

  34. Wang, L., Xu, L., Zhao, H.: Phys. Rev. E 91, 012110 (2015)

    Article  ADS  Google Scholar 

  35. Ernst, M.H., Hauge, E.H., van Leeuwen, J.M.J.: Phys. Rev. A 4, 2055 (1971)

    Article  ADS  Google Scholar 

  36. Ernst, M., Hauge, E., van Leeuwen, J.: J. Stat. Phys. 15(1), 7 (1976)

    Article  ADS  Google Scholar 

  37. Ernst, M., Hauge, E., van Leeuwen, J.: J. Stat. Phys. 15(1), 23 (1976)

    Article  ADS  Google Scholar 

  38. Lippi, A., Livi, R.: J. Stat. Phys. 100, 1147 (2000)

    Article  Google Scholar 

  39. Yang, L., Grassberger, P., Hu, B.: Phys. Rev. E 74, 062101 (2006)

    Article  ADS  Google Scholar 

  40. Xiong, D., Wang, J., Zhang, Y., Zhao, H.: Phys. Rev. E 82, 030101 (2010)

    Article  ADS  Google Scholar 

  41. Grassberger, P., Yang, L.: eprint arXiv:cond-mat/0204247

    Google Scholar 

  42. Chang, C.W., Okawa, D., Garcia, H., Majumdar, A., Zettl, A.: Phys. Rev. Lett. 101, 075903 (2008)

    Article  ADS  Google Scholar 

  43. Ghosh, S., Bao, W., Nika, D.L., Subrina, S., Pokatilov, E.P., Lau, C.N., Balandin, A.A.: Nat. Mater. 9, 555 (2010)

    Article  ADS  Google Scholar 

  44. Xu, X., Wang, Y., Zhang, K., Zhao, X., Bae, S., Heinrich, M., Tinh Bui C., Xie, R., Thong, J.T.L., Hong, B.H., Loh, K.P., Li, B., Oezyilmaz, B.: eprint arXiv:1012.2937

    Google Scholar 

  45. Wang, Z., Xie, R., Bui, C.T., Liu, D., Ni, X., Li, B., Thong, J.T.L.: Nano Lett. 11, 113 (2011)

    Article  ADS  Google Scholar 

  46. Nika, D.L., Askerov, A.S., Balandin, A.A.: Nano Lett. 12, 3238 (2012)

    Article  Google Scholar 

  47. Shiba, H., Yukawa, S., Ito, N.: J. Phys. Soc. Jpn. 75(10), 103001 (2006)

    Article  ADS  Google Scholar 

  48. Shiba, H., Ito, N.: J. Phys. Soc. Jpn. 77(5), 054006 (2008)

    Article  ADS  Google Scholar 

  49. Saito, K., Dhar, A.: Phys. Rev. Lett. 104, 040601 (2010)

    Article  ADS  Google Scholar 

  50. Lepri, S., Livi, R., Politi, A.: Europhys. Lett. 43, 271 (1998)

    Article  ADS  Google Scholar 

  51. Lepri, S., Livi, R., Politi, A.: Phys. Rev. E 68, 067102 (2003)

    Article  ADS  Google Scholar 

  52. Morales, A.M., Lieber, C.M.: Science 279(5348), 208 (1998)

    Article  ADS  Google Scholar 

  53. Ma, D.D.D., Lee, C.S., Au, F.C.K., Tong, S.Y., Lee, S.T.: Science 299(5614), 1874 (2003)

    Article  ADS  Google Scholar 

  54. Zhao, H.: Phys. Rev. Lett. 96, 140602 (2006)

    Article  ADS  Google Scholar 

  55. Liu, S., Hänggi, P., Li, N., Ren, J., Li, B.: Phys. Rev. Lett. 112, 040601 (2014)

    Article  ADS  Google Scholar 

  56. Cipriani, P., Denisov, S., Politi, A.: Phys. Rev. Lett. 94, 244301 (2005)

    Article  ADS  Google Scholar 

  57. Wang, L., Wu, Z., Xu, L.: Phys. Rev. E 91, 062130 (2015)

    Article  ADS  Google Scholar 

  58. Giardinà, C., Livi, R., Politi, A., Vassalli, M.: Phys. Rev. Lett. 84, 2144 (2000)

    Article  ADS  Google Scholar 

  59. Li, Y., Liu, S., Li, N., Hänggi, P., Li, B.: New J. Phys. 17, 043064 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  60. Li, N., Li, B., Flach, S.: Phys. Rev. Lett. 105, 054102 (2010)

    Article  ADS  Google Scholar 

  61. Hu, B., Li, B., Zhao, H.: Phys. Rev. E 61, 3828 (2000)

    Article  ADS  Google Scholar 

  62. Aoki, K., Kusnezov, D.: Phys. Lett. A 265(4), 250 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  63. Skokos, C., Krimer, D.O., Komineas, S., Flach, S.: Phys. Rev. E 79, 056211 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  64. Laskar, J., Robutel, P.: Celest. Mech. Dyn. Astron. 80(1), 39 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  65. Lee-Dadswell, G.R., Nickel, B.G., Gray, C.G.: Phys. Rev. E 72, 031202 (2005)

    Article  ADS  Google Scholar 

  66. Zhao, H., Wen, Z., Zhang, Y., Zheng, D.: Phys. Rev. Lett. 94, 025507 (2005)

    Article  ADS  Google Scholar 

  67. Helfand, E.: Phys. Rev. 119, 1 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  68. Resibois, P., de Leener, M.: Classical Kinetic Theory of Fluids. Wiley, New York/London/Sidney/Toronto (1977)

    MATH  Google Scholar 

  69. Spohn, H.: J. Stat. Phys. 154, 1191 (2014)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 11275267(L.W.), Nos. 11334007 and 11205114 (N.L.), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China 15XNLQ03 (L.W.), the Program for New Century Excellent Talents of the Ministry of Education of China with Grant No. NCET-12-0409 (N.L.), the Shanghai Rising-Star Program with grant No. 13QA1403600 (N.L.). Computational resources were provided by the Physical Laboratory of High Performance Computing at Renmin University of China(L.W.) and Shanghai Supercomputer Center (N.L.).

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Authors and Affiliations

  1. Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing, 100872, People’s Republic of China

    Lei Wang

  2. Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai, 200092, People’s Republic of China

    Nianbei Li

  3. Institute of Physics, University of Augsburg, Augsburg, 86135, Germany

    Peter Hänggi

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  1. Lei Wang
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  2. Nianbei Li
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  3. Peter Hänggi
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Correspondence to Lei Wang .

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Editors and Affiliations

  1. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Sesto Fiorentino (FI), Italy

    Stefano Lepri

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Wang, L., Li, N., Hänggi, P. (2016). Simulation of Heat Transport in Low-Dimensional Oscillator Lattices. In: Lepri, S. (eds) Thermal Transport in Low Dimensions. Lecture Notes in Physics, vol 921. Springer, Cham. https://doi.org/10.1007/978-3-319-29261-8_6

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