Oscillatory-damping temperature behavior in one-dimensional harmonic model of a perfect crystal |

Guzev M.A., Dmitriev A.A. |

2017, issue 2, Ñ. 170-179 |

Abstract |

We constructed an analytical solution for the equations modeling a one-dimensional harmonic crystal. The solution is used to calculate the temperature as a measure of kinetic energy. For stochastic initial conditions, we obtain a law of temperature distribution which differs from the Fourier law. It is demonstrated that the correlations linking the position of the particles leads to the appearance of harmonics at twice the frequency compared with the main oscillation generated due to correlations between the initial velocities. |

one-dimensional harmonic crystal, the temperature distribution, correlation, speedKeywords: |

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## References |

[1] Chandrasekharaiah D.S., “Thermo-elasticity with second sound”, Appl. Mech. Rev., 39:3, (1986), 355–376. [2] Rieder Z., Lebowitz J.L., Lieb E., “Properties of a Harmonic Crystal in a Stationary Nonequilibrium State”, J. Math. Phys, 8:5, (1967), 1073–1078. [3] Cattaneo C., “Sulla conduzione de calore”, Atti Semin. Mat. Fis. Univ. Modena, 3, (1948), 3. [4] Cattaneo C., “A form of heat conduction equation which eliminates the paradox of instantaneous propagation”, Comptes Rendus, 247, (1958), 431. [5] Vernotte P., “Les paradoxes de la theorie continue de l’equation de la chaleur”, Comptes Rendus, 256, (1958), 3154. [6] Vernotte P., “Some possible complications in the phenomena of thermal conduction”, Comptes Rendus, 252, (1961), 2190. [7] Sadd M.H., Didlake J.E., “Non-Fourier Melting of a Semi-Infinite Solid”, J. Heat Transfer, 99:1, (1977), 25–28. [8] Abdulmuhsen Ali, “Statistical Mechanical Derivation of Cattaneo’s Heat Flux Law”, Journal of Thermophysics and Heat Transfer, 13:4, (1999), 544–545. [9] Bahrami A., Hoseinzadeh S., Ghasemiasl R., “Solution of Non-Fourier Temperature Field in a Hollow Sphere under Harmonic Boundary Condition”, Applied Mechanics and Materials, 772, (2015), 197–203. [10] Polyanin A.D., Vyazmin A.V., “Differential-difference heat-conduction and diffusion models and equations with a finite relaxation time”, Theoretical Foundations of Chemical Engineering, 47:3, (2013), 217–224. [11] Krivtsov A.M., “Rasprostranenie tepla v beskonechnom garmonicheskom kristalle”, DAN, 464:2, (2015), 162–166. [12] Guzev M.A., Dmitriev A.A., “Razlichnye formy predstavleniia resheniia odnomernoi garmonicheskoi modeli kristalla”, Dal'nevostochnyi matem. zhurnal, 17:1, (2017), 30–47. [13] Fedoriuk M.V., Asimptorika: integraly i riady (SMB), Nauka: GRFML, Moskva, 1987. |