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| − | [[Виртуальная лаборатория|Virtual laborotory]] > [[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]], published at [http://arxiv.org/abs/1509.02506 arXiv:1509.02506 (cond-mat.stat-mech)]
| + | * [[: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: periodic temperature]] |
| − | Programming: [[Д.В. Цветков|D.V. Tsvetkov]]
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| − | == Microscopic model ==
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| − | We consider a one-dimensional crystal, described by the following equations of motion:
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| − | :<math>
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| − | \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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| − | == Simulation: evolution of the spatial distribution of the kinetic temperature ==
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| − | {{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Tcvetkov/Equations/Equation%20v8b-10%20debug%20random/Equations.html |width=1030 |height=785 |border=0 }}
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| − | == Kinetic temperature: link between micro and macro ==
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| − | The kinetic temperature <math>T</math> is defined as
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| − | :<math>
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| − | T(x) = \frac m{k_{B}}\langle\dot u_i^2\rangle,
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| − | </math>
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| − | where
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| − | <math>k_{B}</math> is the Boltzmann constant,
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| − | <math>i=x/a</math>,
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| − | angle brackets stand for mathematical expectation.
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| − | == Macroscopic equations ==
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| − | {{oncolor||red|—}} Heat (Fourier): <math>\dot T = \beta T''</math> [https://en.wikipedia.org/wiki/Heat_equation]
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| − | {{oncolor||#008888|—}} Heat wave (MCV): <math>\ddot T +\frac1\tau\dot T = \frac\beta\tau T''</math>
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| − | {{oncolor||#00ff00|—}} Wave (d’Alembert): <math>\ddot T = c^2 T''</math> [https://en.wikipedia.org/wiki/Wave_equation]
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| − | {{oncolor||blue|—}} Reversible (Krivtsov): <math>\ddot T +\frac1t\dot T = c^2 T''</math> [http://arxiv.org/abs/1509.02506]
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| − | Notations:
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| − | <math>t</math> is time (variable),
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| − | <math>\tau</math> is the relaxation time (constant),
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| − | <math>\beta</math> is the thermal diffusivity,
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| − | <math>\kappa</math> is the thermal conductivity,
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| − | <math>c</math> is the sound speed,
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| − | <math>\rho</math> is the density,
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| − | MCV stands for Maxwell-Cattaneo-Vernotte.
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| − | == See also ==
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| − | * [[Heat transfer in a 1D harmonic crystal: periodic temperature]]
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| − | * [[Heat transfer in a 1D harmonic crystal: regular temperature]] | |
