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

Abstract

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.

Published manuscript with some small misprints corrected

References in corpus (9)

• Length-dependent thermal conductivity in suspended single-layer graphene
• Heat Transport in low-dimensional systems
• Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs
• Anomalous Heat Conduction and Anomalous Diffusion in Low Dimensional Nanoscale Systems
• Fast and slow thermal processes in harmonic scalar lattices
• New analytic solution for the heat flow through a general harmonic network
• Heat conduction and phonon localization in disordered harmonic crystals
• Harmonic chain with velocity flips: thermalization and kinetic theory
• Heat conduction in a chain of colliding particles with stiff repulsive potential

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• Steady-state ballistic thermal transport associated with transversal motions in a damped graphene lattice subjected to a point heat source
• Thermal equilibration in a one-dimensional damped harmonic crystal
• Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation
• Discrete and continuum fundamental solutions describing heat conduction in 1D harmonic crystal: Discrete-to-continuum limit and slow-and-fast motions decoupling
• Non-stationary elastic wave scattering and energy transport in a one-dimensional harmonic chain with an isotopic defect
• Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain
• Anti-localization of non-stationary quasi-waves in a strongly nonlinear $\beta$-FPUT chain