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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
R. V. Goldshtein and N. F. Morozov, Phys. Mesomech. 10 (5), 17 (2007).
A. M. Krivtsov, Dokl. Phys. 59 (9) (2014).
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].
M. B. Babenkov, A. M. Krivtsov, and D. V. Tsvetkov, Phys. Mesomech. 19 (1), 60 (2016).
C. F. Petersen, D. J. Evans, and S. R. Williams, J. Chem. Phys. 144, 074107 (2016).
F. Silva, S. M. Teichmann, S. L. Cousin, M. Hemmer, and J. Biegert, Nat. Commun. 6, 6611 (2015).
B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, Phys. Rev. A 22, 2798 (1980).
M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987).
Z. Rieder, J. L. Lebowitz, and E. Lieb, J. Math. Phys. 8 (5), 1073 (1967).
A. M. Krivtsov, Dokl. Phys. 60 (9), 407 (2015).
A. M. Krivtsov, ArXiv:1509.02506 [cond-mat.stat-mech] (2015).
G. Benettin and A. Tenenbaum, Phys. Rev. A 28, 3020 (1983).
S. Lepri, R. Livi, and A. Politi, Phys. Rep. 377 (1), 1 (2003).
A. Dhar, Adv. Phys. 57 (5), 457 (2008).
A. V. Savin and O. V. Gendelman, Phys. Rev. E 67 (4), 041205 (2003).
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.
About this article
Cite this article
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