Редактирование: Heat transfer in a 1D harmonic crystal

[[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.