Heat transfer in a 1D harmonic crystal — различия между версиями

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[[Виртуальная лаборатория]] > [[Heat transfer in a 1D harmonic crystal]] <HR>
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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]]
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Theory: [[А.М. Кривцов|A.M. Krivtsov]]
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* [[: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]]
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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>
 
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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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