Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 255–272 | Cite as

Heat transfer in a one-dimensional harmonic crystal in a viscous environment subjected to an external heat supply

  • S. N. GavrilovEmail author
  • A. M. Krivtsov
  • D. V. Tsvetkov
Original Article


We consider unsteady heat transfer in a one-dimensional harmonic crystal surrounded by a viscous environment and subjected to an external heat supply. The basic equations for the crystal particles are stated in the form of a system of stochastic differential equations. We perform a continualization procedure and derive an infinite set of linear partial differential equations for covariance variables. An exact analytic solution describing unsteady ballistic heat transfer in the crystal is obtained. It is shown that the stationary spatial profile of the kinetic temperature caused by a point source of heat supply of constant intensity is described by the Macdonald function of zero order. A comparison with the results obtained in the framework of the classical heat equation is presented. We expect that the results obtained in the paper can be verified by experiments with laser excitation of low-dimensional nanostructures.


Ballistic heat transfer Harmonic crystal Kinetic temperature 


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

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

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