Редактирование: Heat transfer in a 1D harmonic crystal
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− | + | [[Виртуальная лаборатория|Virtual laborotory]] > [[Heat transfer in a 1D harmonic crystal]] <HR> | |
− | + | ||
− | + | Theory: [[А.М. Кривцов|A.M. Krivtsov]], published at [http://arxiv.org/abs/1509.02506 arXiv:1509.02506 (cond-mat.stat-mech)] | |
− | + | ||
+ | Programming: [[Д.В. Цветков|D.V. Tsvetkov]] | ||
+ | |||
+ | == Microscopic model == | ||
+ | |||
+ | We consider a one-dimensional crystal, described by the following equations of motion: | ||
+ | :<math> | ||
+ | \ddot{u}_i = \omega_0^2(u_{i-1}-2u_i+u_{i+1}) | ||
+ | ,\qquad \omega_0 = \sqrt{C/m}, | ||
+ | </math> | ||
+ | where | ||
+ | <math>u_i</math> is the displacement of the <math>i</math>th particle, | ||
+ | <math>m</math> is the particle mass, | ||
+ | <math>C</math> is the stiffness of the interparticle bond. | ||
+ | The crystal is infinite: the index <math>i</math> is an arbitrary integer. | ||
+ | The initial conditions are | ||
+ | :<math> | ||
+ | u_i|_{t=0} = 0 | ||
+ | ,\qquad | ||
+ | \dot u_i|_{t=0} = \sigma(x)\varrho_i | ||
+ | , | ||
+ | </math> | ||
+ | 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. | ||
+ | |||
+ | == Simulation: evolution of the spatial distribution of the kinetic temperature == | ||
+ | |||
+ | {{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Tcvetkov/Equations/Equation%20v8b-8%20debug%206eq%20non_rnd_energy%20ENG/Equations.html |width=1030 |height=785 |border=0 }} | ||
+ | |||
+ | == Kinetic temperature: link between micro and macro == | ||
+ | |||
+ | The kinetic temperature <math>T</math> is defined as | ||
+ | :<math> | ||
+ | T(x) = \frac m{k_{B}}\langle\dot u_i^2\rangle, | ||
+ | </math> | ||
+ | where | ||
+ | <math>k_{B}</math> is the Boltzmann constant, | ||
+ | <math>i=x/a</math>, | ||
+ | angle brackets stand for mathematical expectation. | ||
+ | |||
+ | == Macroscopic equations == | ||
+ | |||
+ | {{oncolor||red|—}} Heat (Fourier): <math>\dot T = \beta T''</math> [https://en.wikipedia.org/wiki/Heat_equation] | ||
+ | |||
+ | {{oncolor||#008888|—}} Heat wave (MCV): <math>\ddot T +\frac1\tau\dot T = \frac\beta\tau T''</math> | ||
+ | |||
+ | {{oncolor||#00ff00|—}} Wave (d’Alembert): <math>\ddot T = c^2 T''</math> [https://en.wikipedia.org/wiki/Wave_equation] | ||
+ | |||
+ | {{oncolor||blue|—}} Reversible (Krivtsov): <math>\ddot T +\frac1t\dot T = c^2 T''</math> [http://arxiv.org/abs/1509.02506] | ||
+ | |||
+ | Notations: | ||
+ | <math>t</math> is time (variable), | ||
+ | <math>\tau</math> is the relaxation time (constant), | ||
+ | <math>\beta</math> is the thermal diffusivity, | ||
+ | <math>\kappa</math> is the thermal conductivity, | ||
+ | <math>c</math> is the sound speed, | ||
+ | <math>\rho</math> is the density, | ||
+ | MCV stands for Maxwell-Cattaneo-Vernotte. |