Localized heat perturbation in harmonic 1D crystals: Solutions for the equation of anomalous heat conduction

Sokolov, A. A.; Кривцов, А. М.; Müller, Wolfgang

doi:10.1134/s1029959917030067

Cited by 27 publications

(15 citation statements)

References 20 publications

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“…Unsteady ballistic heat transport in harmonic crystals is investigated e.g. in papers [4,20,21,23,36,43,48,61]. In paper [36], an equation, refereed to as the ballistic heat equation, describing evolution of temperature field in a one-dimensional chain with interactions of the nearest neighbors is derived.…”

Section: Introductionmentioning

confidence: 99%

Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

Kuzkin

2019

Continuum Mech. Thermodyn.

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We study thermal processes in infinite harmonic crystals having a unit cell with 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 approximate 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 of the temperature profile caused by ballistic heat transport. It is shown, in particular, that local values of temperatures, corresponding to degrees of freedom of the unit cell, are generally different even if their initial values are equal. Analytical findings are supported by results of numerical solution of lattice dynamics equations for diatomic chain and graphene lattice. 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 temperature profile can be used for proper interpretation of experimental data obtained by the transient thermal grating technique.

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Section: Introductionmentioning

confidence: 99%

Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

Kuzkin

2019

Continuum Mech. Thermodyn.

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“…In particular, a heat transfer equation was obtained that differs from the extended heat transfer equations suggested earlier [40][41][42][43]; however, it is in an excellent agreement with molecular dynamics simulations and previous analytical estimates [31]. The properties of the solutions describing heat transfer in a one-dimensional harmonic crystal were discussed in [44][45][46]. Later this approach was generalized [37,38,[47][48][49][50][51]] to a number of systems, namely, to an infinite one-dimensional crystal on an elastic substrate [47], to an infinite one-dimensional diatomic harmonic crystal [52], to a finite one-dimensional crystal [51], and to two and three-dimensional infinite harmonic lattices [48][49][50].…”

mentioning

confidence: 66%

“…Later this approach was generalized [37,38,[47][48][49][50][51]] to a number of systems, namely, to an infinite one-dimensional crystal on an elastic substrate [47], to an infinite one-dimensional diatomic harmonic crystal [52], to a finite one-dimensional crystal [51], and to two and three-dimensional infinite harmonic lattices [48][49][50]. In the most of the above mentioned papers [37,38,44,[47][48][49][50][51] only isolated systems were considered.…”

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confidence: 99%

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

Гаврилов

Кривцов

2019

Continuum Mech. Thermodyn.

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We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite system of linear partial differential equations for covariance variables. The most important results of the paper are the deterministic differential-difference equation describing non-stationary heat propagation in the lattice and the analytical formula in the integral form for its steady-state solution describing kinetic temperature distribution caused by a point heat source of a constant intensity. The comparison between numerical solution of stochastic equations and obtained analytical solution demonstrates a very good agreement everywhere except for the main diagonals of the lattice (with respect to the point source position), where the analytical solution is singular.Keywords ballistic heat transfer · 2D harmonic scalar lattice · kinetic temperature 1 Introduction At the macroscale, Fourier's law of heat conduction is widely and successfully used to describe heat transfer processes. However, recent experimental observations demonstrate that Fourier's law is violated at the microscale and nanoscale, in particular, in low-dimensional nanostructures [1][2][3][4][5][6], where the ballistic heat transfer is realized. The anomalous heat transfer also may be related with the spontaneous emergence of long-range correlations; the latter is typical for momentum-conserving systems [7,8]. The simplest theoretical approach to describe the ballistic heat propagation is to use harmonic lattice models. In some cases such models allow one to obtain the analytical description of thermomechanical processes in solids [9][10][11][12][13][14][15]. In the literature, the problems concerning heat transfer in harmonic lattices are mostly considered

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“…2π 0 θ 0 (x + ct cos p) dp (14) and solutions of the particular initial problems [26]. We note that the exact solution for the atomic velocities and displacements (see Refs.…”

Section: B Hyperbolic Equation (Maxwell-cattaneo-vernotte Type):tmentioning

confidence: 99%

“…The mathematical properties of the ballistic heat equation were investigated in several papers including Ref. [26]. The ballistic heat equation is reversible with respect to a substitution of t to −t.…”

Section: Introductionmentioning

confidence: 99%

Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation

et al. 2019

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This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). A formula for calculating the entropy within the framework of EIT for the ballistic heat equation is derived. The entropy is calculated for a sinusoidal initial temperature perturbation by using both approaches. The results obtained from CIT show that the entropy is a non-monotonic function and that the entropy production can be negative. The results obtained for EIT show that the entropy is a monotonic function and that the entropy production is nonnegative. A comparison between the entropy behaviors predicted for the ballistic, for the ordinary Fourier-based, and for the hyperbolic heat equation is made. A crucial difference of the asymptotic behavior of the entropy for the ballistic heat equation is shown. It is argued that mathematical time reversibility of the partial differential ballistic heat equation is not consistent with its physical irreversibility. The processes described by the ballistic heat equation are irreversible because of the entropy increase. *

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Localized heat perturbation in harmonic 1D crystals: Solutions for the equation of anomalous heat conduction

Cited by 27 publications

References 20 publications

Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation

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