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| − | [[Виртуальная лаборатория]] > [[Heat transfer in a 1D harmonic crystal]] <HR>
| + | This is an old version of the page, please see the new versions: |
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| − | | + | * [[:en:Heat transfer in a 1D harmonic crystal|Heat transfer in a 1D harmonic crystal]] |
| − | Theory: [[А.М. Кривцов|A.M. Krivtsov]]
| + | * [[:en:Heat transfer in a 1D harmonic crystal: periodic temperature|Heat transfer in a 1D harmonic crystal: periodic temperature]] |
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| + | * [[:en:Heat transfer in a 1D harmonic crystal: regular temperature|Heat transfer in a 1D harmonic crystal: regular temperature]] |
| − | Programming: [[Д.В. Цветков|D.V. Tsvetkov]]
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| − | == Model ==
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| − | We consider a one-dimensional crystal, described by the following equations of motion:
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| − | :<math> | |
| − | \ddot{u}_i = \omega_0^2(u_{i-1}-2u_i+u_{i+1})
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| − | ,\qquad \omega_0 = \sqrt{C/m},
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| − | </math>
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| − | where
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| − | <math>u_i</math> is the displacement of the <math>i</math>th particle,
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| − | <math>m</math> is the particle mass,
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| − | <math>C</math> is the stiffness of the interparticle bond.
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| − | The crystal is infinite: the index <math>i</math> is an arbitrary integer.
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| − | The initial conditions are
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| − | :<math>
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| − | u_i|_{t=0} = 0
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| − | ,\qquad
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| − | \dot u_i|_{t=0} = \sigma(x)\varrho_i
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| − | ,
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| − | </math>
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| − | where <math>\varrho_i</math> are independent random values with zero expectation and unit variance; <math>\sigma</math> is variance of the initial velocities of the particles, which is a slowly varying function of the spatial coordinate <math>x=ia</math>, where <math>a</math> is the lattice constant. These initial conditions correspond to an instantaneous temperature perturbation, which can be induced in crystals, for example, by an ultrashort laser pulse.
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| − | == Macroscopic equations ==
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| − | | |
| − | — Heat (Fourier): <math>\dot T = \beta T''</math>
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| − | — Heat wave (MCV): <math>\ddot T +\frac1\tau\dot T = \frac\beta\tau T''</math>
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| − | — Wave (d’Alembert): <math>\ddot T = c^2 T''</math>
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| − | | |
| − | — Reversible (Krivtsov): <math>\ddot T +\frac1t\dot T = c^2 T''</math>
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