High-frequency thermal processes in harmonic crystals


We consider two high frequency thermal processes in uniformly heated harmonic crystals relaxing towards equilibrium: (i) equilibration of kinetic and potential energies and (ii) redistribution of energy among spatial directions. Equation describing these processes with deterministic initial conditions is derived. Solution of the equation shows that characteristic time of these processes is of the order of ten periods of atomic vibrations. After that time the system practically reaches the stationary state. It is shown analytically that in harmonic crystals temperature tensor is not isotropic even in the stationary state. As an example, harmonic triangular lattice is considered. Simple formula relating the stationary value of the temperature tensor and initial conditions is derived. The function describing equilibration of kinetic and potential energies is obtained. It is shown that the difference between the energies (Lagrangian) oscillates around zero. Amplitude of these oscillations decays inversely proportional to time. Analytical results are in a good agreement with numerical simulations.

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  1. 1.

    R. V. Goldshtein and N. F. Morozov, Phys. Mesomech. 10 (5), 17 (2007).

    Google Scholar 

  2. 2.

    A. M. Krivtsov, Dokl. Phys. 59 (9) (2014).

  3. 3.

    A. M. Krivtsov, Dynamics of Thermal Processes in One- Dimensional Harmonic Crystals. Questions of Methematical Physics and Applied Mathematics (Ioffe Physical- Technical Institute, St. Petersburg, 2016) [in Russian].

    Google Scholar 

  4. 4.

    M. B. Babenkov, A. M. Krivtsov, and D. V. Tsvetkov, Phys. Mesomech. 19 (1), 60 (2016).

    Google Scholar 

  5. 5.

    C. F. Petersen, D. J. Evans, and S. R. Williams, J. Chem. Phys. 144, 074107 (2016).

    ADS  Article  Google Scholar 

  6. 6.

    F. Silva, S. M. Teichmann, S. L. Cousin, M. Hemmer, and J. Biegert, Nat. Commun. 6, 6611 (2015).

    ADS  Article  Google Scholar 

  7. 7.

    B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, Phys. Rev. A 22, 2798 (1980).

    ADS  Article  Google Scholar 

  8. 8.

    M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987).

    Google Scholar 

  9. 9.

    Z. Rieder, J. L. Lebowitz, and E. Lieb, J. Math. Phys. 8 (5), 1073 (1967).

    ADS  Article  Google Scholar 

  10. 10.

    A. M. Krivtsov, Dokl. Phys. 60 (9), 407 (2015).

    ADS  Article  Google Scholar 

  11. 11.

    A. M. Krivtsov, ArXiv:1509.02506 [cond-mat.stat-mech] (2015).

  12. 12.

    G. Benettin and A. Tenenbaum, Phys. Rev. A 28, 3020 (1983).

    ADS  Article  Google Scholar 

  13. 13.

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

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

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

    ADS  Article  Google Scholar 

  15. 15.

    A. V. Savin and O. V. Gendelman, Phys. Rev. E 67 (4), 041205 (2003).

    ADS  Article  Google Scholar 

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Correspondence to A. M. Krivtsov.

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Original Russian Text © V.A. Kuzkin, A.M. Krivtsov, 2017, published in Doklady Akademii Nauk, 2017, Vol. 472, No. 5, pp. 529–533.

The article was translated by the authors.

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Kuzkin, V.A., Krivtsov, A.M. High-frequency thermal processes in harmonic crystals. Dokl. Phys. 62, 85–89 (2017). https://doi.org/10.1134/S1028335817020070

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