The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal

Кривцов, А. М.

doi:10.1007/978-3-030-11665-1_19

Cited by 17 publications

(17 citation statements)

References 44 publications

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“…Remark. Heat transport in several scalar lattices with initial temperature distribution (38) is considered in papers [36,37,43]. In the present paper, we derive a general solution valid for any lattice described by equations of motion (4).…”

Section: Sinusoidal Initial Temperature Profile: Application To Transmentioning

confidence: 99%

“…Since initial conditions (5), (6) are random then particle velocities given by formula (16) are also random. Following [35,36,37] we consider an infinite number of realizations of initial conditions (5), (6). It allows to introduce statistical characteristics such as kinetic temperature.…”

Section: Kinetic Temperature Temperature Matrixmentioning

confidence: 99%

“…Analytical solution of this problem is given by formula (37). To check formula (37), we compare them with results of numerical solution of equations of motion. Formula (37) shows that at large times the solution is self-similar.…”

Section: Thermal Contact Of Cold and Hot Half-planesmentioning

confidence: 99%

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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: Sinusoidal Initial Temperature Profile: Application To Transmentioning

confidence: 99%

Section: Kinetic Temperature Temperature Matrixmentioning

confidence: 99%

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Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

Kuzkin

2019

Continuum Mech. Thermodyn.

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“…We assume that mechanical vibrations of the chain are described by the equation of linear thermoelasticity [26], while behavior of temperature is described by the ballistic heat equation [18,19]. Conversion of mechanical energy to thermal energy is neglected.…”

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confidence: 99%

Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature

Kuzkin

Кривцов

2020

Phys. Rev. E

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We study conversion of thermal energy to mechanical energy and vice versa in α-FPU chain with spatially sinusoidal profile of initial temperature. We show analytically that thermal expansion and temperature oscillations, caused by quasiballistic heat transport, excite mechanical vibrations with growing amplitude. This new phenomenon is referred to as "ballistic resonance". At large times, the mechanical vibrations decay monotonically. No recurrence, typical for the original FPU problem, is observed. We assume that the absence of recurrence is due to finite initial temperature of the chain.Energy of an isolated solid body can be converted from a mechanical form to a thermal form and vice versa. Conversion of mechanical energy to thermal energy leads, for example, to damping of mechanical vibrations (or waves). Thermal expansion and heat transport cause an opposite process, i.e. conversion of thermal energy to mechanical energy.In macroscopic continuum theories, the conversion is modeled by coupling between the equation of momentum balance and the equation of energy balance. An example of a continuum theory, describing the conversion, is the linear thermoviscoelasticity. Macroscopic theory of linear thermoviscoelasticity is well-developed [1]. However at micro-and nanoscale this theory faces significant difficulties, because macroscopic constitutive relations can be inapplicable at lower length scales. For example, recent theoretical [2-4] and experimental studies [5][6][7][8][9] show that at micro-and nanoscale the Fourier law of heat conduction can be violated. In particular, ballistic regime of heat transport is observed [10,11].A convenient model for investigation of thermomechanical processes at micro-and nanoscale is the Fermi-Pasta-Ulam (FPU) chain [12]. Despite the apparent simplicity of the model, analytical description of thermoelasticity, heat transport and energy conversion in the FPU chain remains a serious challenge for theoreticians.Several anomalies of thermomechanical behavior are observed for the FPU chain. The heat transport in the FPU model is anomalous, i.e. the Fourier law is not satisfied [13][14][15]. The Maxwell-Cattaneo-Vernotte law also fails to describe the heat transport in FPU chains [16,17]. Harmonic approximation allows to derive equations [18,19] and closed-form solutions [20, 21] describing heat transport in chains. However the question arises wether temperature field, obtained in harmonic approximation, can be used for estimation of thermoelastic effects (e.g. excitation of mechanical vibrations due to thermal expansion). We address this question below.Conversion of mechanical energy to thermal energy is even more challenging. Studies of this process have a long history, starting from the pioneering work of Fermi, Pasta, and Ulam [12]. In paper [12], initial conditions, corresponding to the first normal mode of a chain, were considered. It was shown numerically that energy of this normal mode demonstrates almost periodic behavior, i.e. the system does not reach thermal equilibrium...

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“…In previous studies [37][38][39], a new approach was suggested which allows one to solve analytically non-stationary thermal problems for an infinite one-dimensional harmonic crystal -an infinite ordered chain of identical material particles, interacting via linear (harmonic) forces. 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].…”

mentioning

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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The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal

Cited by 17 publications

References 44 publications

Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell

Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature

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

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