Difference between revisions of "Nosé–Hoover thermostat"

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== Description ==  
+
== Description of the model ==  
  
Nosé–Hoover thermostat is used to keep the temperature constant in the system. Thermostat is given by:
+
Nosé–Hoover thermostat is used to keep the temperature constant in the system. Equations of motion of the thermostated harmonic oscillator have the form:
  
 
::<math>
 
::<math>
 
\left\{  
 
\left\{  
 
\begin{array}{ll}
 
\begin{array}{ll}
v' =\omega^2_{\rm 0} x - \gamma v \\
+
\dot{v} =\omega^2_{\rm 0} x - \gamma v \\
\displaystyle \gamma' = \frac{1}{\tau^2} \left( \frac{T}{T_{\rm 0}} - 1\right)\\
+
\displaystyle \dot{\gamma} = \frac{1}{\tau^2} \left( \frac{T}{T_{\rm 0}} - 1\right)\\
 
\end{array}
 
\end{array}
 
\right.
 
\right.
 
</math>
 
</math>
  
где
+
where
  
* <math> {\omega}_{\rm 0} = \sqrt{ \frac{c}{m}} </math> - frequency
+
* <math> {\omega}_{\rm 0} = \sqrt{ \frac{c}{m}} </math> is the eigen frequency
  
* <math> {T_{\rm 0}} </math> - the initial temperature of the system
+
* <math> {T_{\rm 0}} </math> is the initial kinetic temperature of the system
  
* <math> {T} </math> - temperature of the system at the current time
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* <math> {T} </math> is the current kinetic temperature of the system  
  
* <math> {v} </math> - speed of body
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* <math> {v} </math> is the  velocity
  
* <math> {\tau} </math> - parameter of the thermostat
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* <math> {\tau} </math> is the relaxation time
  
 
* <math> {tau}_{\rm 0} = 1 </math> -  scale for <math> {\tau} </math>
 
* <math> {tau}_{\rm 0} = 1 </math> -  scale for <math> {\tau} </math>
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* <math>  {c}_{\rm 0} = 1 </math> - scale of stiffness for <math> {c} </math>
 
* <math>  {c}_{\rm 0} = 1 </math> - scale of stiffness for <math> {c} </math>
  
== Graphics Options ==  
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== Phase-space trajectory of thermostated harmonic oscillator ==  
The graph below has three parameters:
+
The plot shows the trajectory of the thermostated harmonic oscillator in the phase-space. The equations of motion are solved numerically using leap-frog integration scheme.  The followng three parameters can be changed by the user:
  
1) tau =  <math> {\tau} </math> - parameter of the thermostat
+
1) tau =  <math> {\tau} </math> is the relaxation time
  
2) stiff  =  <math> {c} </math> - stiffness of the system
+
2) stiff  =  <math> {c} </math> is the stiffness  
  
3) scale - scale of graph
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3) scale is a scale parameter for a plot
  
'''Last slider - the number of pre-configured experiment.'''  
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'''The last slider allows to choose the number of pre-configured experiment.'''  
  
The graph shows the phase plane - dependence <math> V(x) </math>
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{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Markov/Nose%E2%80%93Hoover%20thermostat/Thermostat_en.html |width=1000 |height=720 |border=0 }}
  
{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Markov/Nose%E2%80%93Hoover%20thermostat/Thermostat.html |width=1000 |height=720 |border=0 }}
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== Authorship ==
 +
 
 +
This stand has been developed by [http://tm.spbstu.ru/Nikolai_Markov Nikolai Markov].
  
 
== References ==
 
== References ==
* [[Виртуальная лаборатория]]
+
 
* [[Курсовые работы по ВМДС: 2014-2015 | Курсовые по дисциплине "Введение в механику дискретных сред"]]
+
* S. Nosé (1984). "A unified formulation of the constant temperature molecular-dynamics methods". J. Chem. Phys. 81 (1): 511–519.
 +
* W.G. Hoover, (1985). "Canonical dynamics: Equilibrium phase-space distributions". Phys. Rev. A, 31 (3): 1695–1697.
 +
* D.J. Evans, B.L. Holian (1985) The Nose–Hoover thermostat. J. Chem. Phys. 83, 4069.
 +
 
 +
== Links ==
 +
* [[Virtual laboratory]]
 +
* [http://williamhoover.info/  William Hoover's Homepage]

Latest revision as of 21:07, 26 October 2015

Virtual laboratory > Nosé–Hoover thermostat


Description of the model[edit]

Nosé–Hoover thermostat is used to keep the temperature constant in the system. Equations of motion of the thermostated harmonic oscillator have the form:

[math] \left\{ \begin{array}{ll} \dot{v} =\omega^2_{\rm 0} x - \gamma v \\ \displaystyle \dot{\gamma} = \frac{1}{\tau^2} \left( \frac{T}{T_{\rm 0}} - 1\right)\\ \end{array} \right. [/math]

where

  • [math] {\omega}_{\rm 0} = \sqrt{ \frac{c}{m}} [/math] is the eigen frequency
  • [math] {T_{\rm 0}} [/math] is the initial kinetic temperature of the system
  • [math] {T} [/math] is the current kinetic temperature of the system
  • [math] {v} [/math] is the velocity
  • [math] {\tau} [/math] is the relaxation time
  • [math] {tau}_{\rm 0} = 1 [/math] - scale for [math] {\tau} [/math]
  • [math] {c}_{\rm 0} = 1 [/math] - scale of stiffness for [math] {c} [/math]

Phase-space trajectory of thermostated harmonic oscillator[edit]

The plot shows the trajectory of the thermostated harmonic oscillator in the phase-space. The equations of motion are solved numerically using leap-frog integration scheme. The followng three parameters can be changed by the user:

1) tau = [math] {\tau} [/math] is the relaxation time

2) stiff = [math] {c} [/math] is the stiffness

3) scale is a scale parameter for a plot

The last slider allows to choose the number of pre-configured experiment.

Authorship[edit]

This stand has been developed by Nikolai Markov.

References[edit]

  • S. Nosé (1984). "A unified formulation of the constant temperature molecular-dynamics methods". J. Chem. Phys. 81 (1): 511–519.
  • W.G. Hoover, (1985). "Canonical dynamics: Equilibrium phase-space distributions". Phys. Rev. A, 31 (3): 1695–1697.
  • D.J. Evans, B.L. Holian (1985) The Nose–Hoover thermostat. J. Chem. Phys. 83, 4069.

Links[edit]