Chin. Phys. Lett.  2021, Vol. 38 Issue (4): 044401    DOI: 10.1088/0256-307X/38/4/044401
FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING APPLICATIONS) |
Approach to Phonon Relaxation Time and Mean Free Path in Nonlinear Lattices
Yue Liu  and Dahai He*
Department of Physics, Xiamen University, Xiamen 361005, China
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Yue Liu  and Dahai He 2021 Chin. Phys. Lett. 38 044401
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Abstract Based on the self-consistent phonon theory, the spectral energy density is calculated by the canonical transformation and the Fourier transformation. Through fitting the spectral energy density by the Lorentzian profile, the phonon frequency as well as the phonon relaxation time is obtained in one-dimensional nonlinear lattices, which is validated in the Fermi–Pasta–Ulam-$\beta$ (FPU-$\beta$) and $\phi^{4}$ lattices at different temperatures. The phonon mean free path is then evaluated in terms of the phonon relaxation time and phonon group velocity. The results show that, in the FPU-$\beta$ lattice, the phonon mean free path as well as the phonon relaxation time displays divergent power-law behavior. The divergent exponent coincides well with that derived from the Peierls–Boltzmann theory at weak anharmonic nonlinearity. The value of the divergent exponent expects a power-law divergent heat conductivity with system size, which violates Fourier's law. For the $\phi^{4}$ lattice, both the phonon relaxation time and mean free path are finite, which ensures normal heat conduction.
Received: 19 December 2020      Published: 06 April 2021
PACS:  63.20.Ry (Anharmonic lattice modes)  
  05.60.Cd (Classical transport)  
  63.20.D  
  66.70.  
Fund: Supported by the National Natural Science Foundation of China (Grant Nos. 12075199 and 11675133).
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http://cpl.iphy.ac.cn/10.1088/0256-307X/38/4/044401       OR      http://cpl.iphy.ac.cn/Y2021/V38/I4/044401
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Yue Liu  and Dahai He
[1]Born M and Huang K 1954 Dynamical Theory of Crystal Lattices (Oxford: Oxford University Press)
[2] Lepri S, Livi R and Politi A 2003 Phys. Rep. 377 1
[3] Dhar A 2008 Adv. Phys. 57 457
[4] Lepri S, Livi R and Politi A 1997 Phys. Rev. Lett. 78 1896
[5] Wang L and Wang T 2011 Europhys. Lett. 93 54002
[6] Chang C W, Okawa D, Garcia H, Majumdar A and Zettl A 2008 Phys. Rev. Lett. 101 075903
[7] Narayan O and Ramaswamy S 2002 Phys. Rev. Lett. 89 200601
[8] Spohn H 2014 J. Stat. Phys. 154 1191
[9] Hu B, Li B and Zhao H 1998 Phys. Rev. E 57 2992
[10] Hu B, Li B and Zhao H 2000 Phys. Rev. E 61 3828
[11] Aoki K and Kusnezov D 2000 Phys. Lett. A 265 250
[12] Rieder Z, Lebowitz J L and Lieb E 1967 J. Math. Phys. 8 1073
[13] Xu Y, Wang J S, Duan W, Gu B L and Li B 2008 Phys. Rev. B 78 224303
[14] Feynman R P and Kleinert H 1986 Phys. Rev. A 34 5080
[15] He D, Buyukdagli S and Hu B 2008 Phys. Rev. E 78 061103
[16] He D, Buyukdagli S and Hu B 2009 Phys. Rev. B 80 104302
[17] Cao X, He D, Zhao H and Hu B 2015 AIP Adv. 5 053203
[18] He D, Thingna J, Wang J S and Li B 2016 Phys. Rev. B 94 155411
[19] He D, Thingna J and Cao J 2018 Phys. Rev. B 97 195437
[20] Alabiso C, Casartelli M and Marenzoni P 1995 J. Stat. Phys. 79 451
[21] Alabiso C and Casartelli M 2001 J. Phys. A 34 1223
[22] Gershgorin B, Lvov Y V and Cai D 2005 Phys. Rev. Lett. 95 264302
[23] Li N, Tong P and Li B 2006 Europhys. Lett. 75 49
[24] Gershgorin B, Lvov Y V and Cai D 2007 Phys. Rev. E 75 046603
[25] Li N, Tong P and Li B 2007 Europhys. Lett. 78 34001
[26] Li N, Li B and Flach S 2010 Phys. Rev. Lett. 105 054102
[27] Li N and Li B 2013 Phys. Rev. E 87 042125
[28] Liu S, Liu J, Hänggi P, Wu C and Li B 2014 Phys. Rev. B 90 174304
[29] Xu L and Wang L 2016 Phys. Rev. E 94 030101
[30] Xu L and Wang L 2017 Phys. Rev. E 95 042138
[31] Thomas J A, Turney J E, Iutzi R M, Amon C H and McGaughey A J H 2010 Phys. Rev. B 81 081411
[32] Feng T L and Ruan X L 2014 J. Nanomater. 2014 206370
[33] Liu Y and He D 2019 Phys. Rev. E 100 052143
[34]Fermi E, Pasta J and Ulam S 1965 Collected Papers of Enrico Fermi ed Segré E (Chicago: University of Chicago Press) vol 2 p 978
[35] Pereverzev A 2003 Phys. Rev. E 68 056124
[36] Nickel B 2007 J. Phys. A 40 1219
[37] Lukkarinen J and Spohn H 2008 Commun. Pure Appl. Math. 61 1753
[38] Hu S, Chen J, Yang N and Li B 2017 Carbon 116 139
[39] Liu Y and He D 2017 Phys. Rev. E 96 062119
[40] Fang J, Qian X, Zhao C Y, Li B and Gu X 2020 Phys. Rev. E 101 022133
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