The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal

  • Anton KrivtsovEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 103)


The analytical model of unsteady ballistic heat transfer in a one-dimensional harmonic crystal is analyzed. A nonlocal temperature is introduced as a generalization of the kinetic temperature. A closed equation determining unsteady thermal processes in terms of the nonlocal temperature is derived. For an instantaneous heat perturbation a time-reversible equation for the kinetic temperature is derived and solved. This equation can be referred as the ballistic heat conduction equation, it is somewhat similar to the hyperbolic heat conduction equation, but it has important differences. The resulting constitutive law for the heat flux in the considered system is obtained. This law significantly differs from Fourier’s law and it predicts a finite velocity of the heat front and independence of the heat flux on the crystal length. The analytical results are confirmed by computer simulations. Further developments of the presented approach are referred.



The author is grateful to O. V. Gendelman, W. G. Hoover, D. A. Indeitsev, M. L. Kachanov, V. A. Kuzkin, S. A. Lurie, N. F. Morozov, and V. F. Zhuravlev for helpful and stimulating discussions; to M. B. Babenkov and D. V. Tsvetkov for additional analysis and computations. This work is supported by the Russian Science Foundation (Grant No. 18-11-00201).


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Authors and Affiliations

  1. 1.Department of Theoretical MechanicsPeter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia
  2. 2.Laboratory of Discrete Models in MechanicsInstitute for Problems in Mechanical Engineering RASSt. PetersburgRussia

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