Nosé–Hoover thermostat

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Virtual laboratory > Nosé–Hoover thermostat

Description of the model

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]


  • [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

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.


This stand has been developed by Nikolai Markov.


  • 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.