Continuum Mechanics and Thermodynamics

, Volume 31, Issue 6, pp 1573–1599 | Cite as

Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

  • Vitaly A. KuzkinEmail author
Original Article


We study thermal processes in infinite harmonic crystals having a unit cell with an arbitrary number of particles. Initially, particles have zero displacements and random velocities, corresponding to some initial temperature profile. Our main goal is to calculate spatial distribution of kinetic temperatures, corresponding to degrees of freedom of the unit cell, at any moment in time. An expression for the temperatures is derived from solution of lattice dynamics equations. It is shown that the temperatures are represented as a sum of two terms. The first term describes high-frequency oscillations of the temperatures caused by local transition to thermal equilibrium at short times. The second term describes slow changes in the temperature profile caused by ballistic heat transport. It is shown that during heat transport, local values of temperatures, corresponding to degrees of freedom of the unit cell, are generally different. Analytical findings are supported by results of numerical solution of lattice dynamics equations for diatomic chain and graphene lattice. Strong anisotropy of ballistic heat transport in graphene is demonstrated. Presented theory may serve for description of unsteady ballistic heat transport in real crystals with low concentration of defects. In particular, solution of the problem with sinusoidal initial temperature profile can be used for proper interpretation of experimental data obtained by the transient thermal grating technique.


Ballistic heat transport Ballistic limit Heat transfer Thermal waves Harmonic crystal Harmonic approximation Polyatomic crystal lattice Complex lattice Kinetic temperature Transient processes Temperature matrix Energy transport Graphene Anisotropy 



The author is deeply grateful to his teacher A.M. Krivtsov for the general statement of the problem and continuous stimulating discussions. The author greatly appreciates discussions with S.V. Gavrilov, M.A. Guzev, S.V. Dmitriev, D.A. Indeitsev, M.L. Kachanov, and A.A. Maznev. The work was financially supported by the Russian Science Foundation under grant No. 18-11-00201.


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Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering RASSaint PetersburgRussia
  2. 2.Peter the Great St. Petersburg Polytechnic UniversitySaint PetersburgRussia

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