Difference between revisions of "Heat transfer in a 1D harmonic crystal"

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[[ru:Распространение тепла в гармоническом одномерном кристалле]]
 
[[ru:Распространение тепла в гармоническом одномерном кристалле]]
 
[[Virtual laboratory]] > [[Heat transfer in a 1D harmonic crystal]] <HR>
 
[[Virtual laboratory]] > [[Heat transfer in a 1D harmonic crystal]] <HR>
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[[A.M. Krivtsov]] (analytical silution, simulation algorithms), [[D.V. Tsvetkov]] (programming, calculation algorithms).<HR>
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__NOEDITSECTION__
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__NOTOC__
  
Theory: [[А.М. Кривцов|A.M. Krivtsov]], published at [http://arxiv.org/abs/1509.02506 arXiv:1509.02506 (cond-mat.stat-mech)]
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Heat transfer in the simplest discrete systems doesn’t obey common macroscopic laws. Recently experimentalists have observed the similar behavior at nanolevel, in molecular and atomic systems. The simulation below demonstrates heat transfer process in a 1D harmonic crystal. Two graphs are shown: results of molecular dynamics simulation and corresponding continuum solution. You can also compare the results with predictions of other continuum models. The analysis of the system and derivation of the continuum solution are presented in paper: ArXiv:1509.02506 ([http://arxiv.org/abs/1509.02506 abstract], [http://arxiv.org/pdf/1509.02506v2.pdf pdf]). See also information below the simulation.  
 
Programming: D.V. Tsvetkov
 
  
== Microscopic model ==
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{{oncolor|yellow|black|Use the '''Restart''' button to see the process from the beginning.}}
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{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Tcvetkov/Equations/Equation%20v8b-10%20debug%20random/Equations.html |width=1030 |height=745 |border=0 }}
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== Discrete model (microlevel) ==
  
 
We consider a one-dimensional crystal, described by the following equations of motion:
 
We consider a one-dimensional crystal, described by the following equations of motion:
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     ,
 
     ,
 
</math>
 
</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.
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where <math>\varrho_i</math> are independent random values with zero expectation and unit variance; <math>\sigma^2(x)</math> is variance of the initial velocities, 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. Periodic conditions are used at the boundaries.
 
 
== Simulation: evolution of the spatial distribution of the kinetic temperature ==
 
  
{{#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) ==
 
 
== Kinetic temperature: link between micro and macro ==
 
  
 
The kinetic temperature <math>T</math> is defined as  
 
The kinetic temperature <math>T</math> is defined as  
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angle brackets stand for mathematical expectation.
 
angle brackets stand for mathematical expectation.
  
== Macroscopic equations ==
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== Continuum description (macrolevel) ==
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{{oncolor||blue|—}} Reversible heat wave equation: <math>\ddot T +\frac1t\dot T = c^2 T''</math> — this equation is derived as a direct consequence of the discrete microscopic equations [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>c</math> is the sound speed.
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== Classic continuum equations ==
  
 
{{oncolor||red|—}} Heat (Fourier): <math>\dot T = \beta T''</math> [https://en.wikipedia.org/wiki/Heat_equation]
 
{{oncolor||red|—}} Heat (Fourier): <math>\dot T = \beta T''</math> [https://en.wikipedia.org/wiki/Heat_equation]
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{{oncolor||#00ff00|—}} Wave (d’Alembert): <math>\ddot T = c^2 T''</math> [https://en.wikipedia.org/wiki/Wave_equation]
 
{{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:
 
Notations:
<math>t</math> is time (variable),
 
 
<math>\tau</math> is the relaxation time (constant),
 
<math>\tau</math> is the relaxation time (constant),
 
<math>\beta</math> is the thermal diffusivity,
 
<math>\beta</math> is the thermal diffusivity,
 
<math>\kappa</math> is the thermal conductivity,
 
<math>\kappa</math> is the thermal conductivity,
<math>c</math> is the sound speed,
 
 
<math>\rho</math> is the density,
 
<math>\rho</math> is the density,
 
MCV stands for Maxwell-Cattaneo-Vernotte.
 
MCV stands for Maxwell-Cattaneo-Vernotte.
  
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== Related publications ==
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* [[A.M. Krivtsov]]. '''On unsteady heat conduction in a harmonic crystal.''' 2015, ArXiv:1509.02506 ([http://arxiv.org/abs/1509.02506 abstract], [http://arxiv.org/pdf/1509.02506v2.pdf pdf])
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* [[A.M. Krivtsov]]. '''Energy oscillations in a one-dimensional crystal.''' Doklady Physics, 2014, Vol. 59, No. 9, pp. 427–430 (pdf: [[Media: Krivtsov_2014_DAN_eng_corrected.pdf| 162 Kb]])
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<!--
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* [[A.A. Le-Zakharov]], [[A.M. Krivtsov]]. '''Molecular dynamics investigation of heat conduction in crystals with defects.'''  Doklady Physics, 2008, Vol. 53, No. 5, pp. 261–264 (pdf: [http://www.ipme.ru/ipme/labs/msm/Pub/Le_2008_DAN_eng.pdf 196 Kb])
  
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* [[A.M. Krivtsov]] '''From nonlinear oscillations to equation of state in simple discrete systems.''' Chaos, Solitons & Fractals, 2003, 17(1), 79-87. (pdf: [http://www.ipme.ru/ipme/labs/msm/Pub/Krivtsov_2003_CSF.pdf 117 Kb])
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-->
  
== See also ==
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== Presentations ==
  
* [[Heat transfer in a 1D harmonic crystal: periodic temperature]]
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* [[A.M. Krivtsov]]. '''One-dimensional crystals and heat superconductivity.''' [http://www.apm-conf.spb.ru International Summer School-Conference “Advanced Problems in Mechanics”], 2015, St. Petersburg, Russia. ''Plenary lecture:'' [[Media: Krivtsov_2015_06_22_APM_09-02_modified_151010_.pdf|pdf 1913 Kb]].
* [[Heat transfer in a 1D harmonic crystal: regular temperature]]
 
  
 
[[Category: Virtual laboratory]]
 
[[Category: Virtual laboratory]]

Latest revision as of 00:04, 13 October 2015

Virtual laboratory > Heat transfer in a 1D harmonic crystal
A.M. Krivtsov (analytical silution, simulation algorithms), D.V. Tsvetkov (programming, calculation algorithms).


Heat transfer in the simplest discrete systems doesn’t obey common macroscopic laws. Recently experimentalists have observed the similar behavior at nanolevel, in molecular and atomic systems. The simulation below demonstrates heat transfer process in a 1D harmonic crystal. Two graphs are shown: results of molecular dynamics simulation and corresponding continuum solution. You can also compare the results with predictions of other continuum models. The analysis of the system and derivation of the continuum solution are presented in paper: ArXiv:1509.02506 (abstract, pdf). See also information below the simulation.

Use the Restart button to see the process from the beginning.

Discrete model (microlevel)

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^2(x)[/math] is variance of the initial velocities, 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. Periodic conditions are used at the boundaries.

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.

Continuum description (macrolevel)

Reversible heat wave equation: [math]\ddot T +\frac1t\dot T = c^2 T''[/math] — this equation is derived as a direct consequence of the discrete microscopic equations [1]

Notations: [math]t[/math] is time (variable), [math]c[/math] is the sound speed.

Classic continuum equations

Heat (Fourier): [math]\dot T = \beta T''[/math] [2]

Heat wave (MCV): [math]\ddot T +\frac1\tau\dot T = \frac\beta\tau T''[/math]

Wave (d’Alembert): [math]\ddot T = c^2 T''[/math] [3]

Notations: [math]\tau[/math] is the relaxation time (constant), [math]\beta[/math] is the thermal diffusivity, [math]\kappa[/math] is the thermal conductivity, [math]\rho[/math] is the density, MCV stands for Maxwell-Cattaneo-Vernotte.

Related publications

  • A.M. Krivtsov. Energy oscillations in a one-dimensional crystal. Doklady Physics, 2014, Vol. 59, No. 9, pp. 427–430 (pdf: 162 Kb)

Presentations