Редактирование: Heat transfer in a 1D harmonic crystal

[[Виртуальная лаборатория]] > [[Heat transfer in a 1D harmonic crystal]] <HR>


Theory: [[А.М. Кривцов|A.M. Krivtsov]]
 
Programming: [[Д.В. Цветков|D.V. Tsvetkov]]

== 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.