Nosé-Hoover thermostat: Difference between revisions
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The '''Nosé-Hoover thermostat'''<ref>[http://dx.doi.org/10.1063/1.447334 Shuichi Nosé "A unified formulation of the constant temperature molecular dynamics methods" , Journal of Chemical Physics '''81''' pp. 511-519 (1984)]</ref> | |||
<ref>[http://dx.doi.org/10.1080/00268978400101201 Shuichi Nosé "A molecular dynamics method for simulations in the canonical ensemble", Molecular Physics '''52''' pp. 255-268 (1984)]</ref> | |||
<ref>[http://dx.doi.org/10.1103/PhysRevA.31.1695 William G. Hoover "Canonical dynamics: Equilibrium phase-space distributions", Physical Review A '''31''' pp. 1695-1697 (1985)]</ref> is a method for controlling the [[temperature]] in a [[molecular dynamics]] simulation. | |||
The Nosé-Hoover [[thermostats |thermostat]] "strives" to reproduce the [[Canonical ensemble |canonical]] phase-space distribution. It does this by modifying the equations of motion to include a non-Newtonian term in order to maintain the total kinetic energy constant. | |||
The modified equation of motion is given by (Ref. 3 Eq. 4) | |||
:<math>\frac{{\mathrm {d}}{\mathbf{v}}(t)}{{\mathrm {d}t}} = \frac{{\mathbf {F}}(t)}{m} -\zeta {\mathbf{v}}(t)</math> | |||
where <math>\zeta</math> is the thermodynamic friction coefficient, given by (Ref. 3 Eq. 5) | |||
:<math>\frac{{\mathrm {d}}\zeta(t)}{{\mathrm {d}t}} = \frac{1}{Q} \left[ \sum m {\mathbf{v}}(t)^2 - (X+1)k_BT \right]</math> | |||
where <math>Q</math> is a parameter that has the dimensions of energy<math>\times</math>(time)<sup>2</sup> and determines the time-scale of the temperature fluctuation and <math>X</math> is the number of degrees of freedom. | |||
==Problems== | |||
The Nosé-Hoover thermostat has problems with [[Ergodic hypothesis |ergodicity]] for small or stiff systems. In order to compensate for this a modification using "chains" has been proposed <ref>[http://dx.doi.org/10.1063/1.463940 Glenn J. Martyna, Michael L. Klein and Mark Tuckerman "Nosé–Hoover chains: The canonical ensemble via continuous dynamics", Journal of Chemical Physics '''97''' pp. 2635- (1992)]</ref>. | |||
==Non-equilibrium== | |||
A version of the Nosé-Hoover thermostat has been developed for [[Non-equilibrium thermodynamics | non-equilibrium]] simulations <ref>[http://dx.doi.org/10.1063/1.2829869 Ben Leimkuhler, Frédéric Legoll and Emad Noorizadeh "A temperature control technique for nonequilibrium molecular simulation", Journal of Chemical Physics '''128''' 074105 (2008)]</ref>. | |||
==References== | ==References== | ||
<references/> | |||
'''Related reading''' | |||
*[http://dx.doi.org/10.1063/1.449071 D. J. Evans and B. L. Holian "The Nose–Hoover thermostat", Journal of Chemical Physics '''83''' pp. 4069-4074 (1985)] | |||
*[http://dx.doi.org/10.1063/1.2013227 Carlos Braga and Karl P. Travis "A configurational temperature Nosé-Hoover thermostat", Journal of Chemical Physics '''123''' 134101 (2005)] | |||
* See http://williamhoover.info and Wm. G. Hoover and Carol G. Hoover "Time Reversibility, Computer Simulations, Algorithms, Chaos", Advanced Series in Nonlinear Dynamics '''13''' World Scientific (2012) ISBN 978-981-4383-16-5 | |||
[[category: molecular dynamics]] | |||
Latest revision as of 12:18, 4 April 2014
The Nosé-Hoover thermostat[1] [2] [3] is a method for controlling the temperature in a molecular dynamics simulation. The Nosé-Hoover thermostat "strives" to reproduce the canonical phase-space distribution. It does this by modifying the equations of motion to include a non-Newtonian term in order to maintain the total kinetic energy constant. The modified equation of motion is given by (Ref. 3 Eq. 4)
where is the thermodynamic friction coefficient, given by (Ref. 3 Eq. 5)
![{\displaystyle {\frac {{\mathrm {d} }\zeta (t)}{\mathrm {d} t}}={\frac {1}{Q}}\left[\sum m{\mathbf {v} }(t)^{2}-(X+1)k_{B}T\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/039cc14a8a6a5471a4c428e173d1287bbf4a4a87)
where is a parameter that has the dimensions of energy(time)2 and determines the time-scale of the temperature fluctuation and is the number of degrees of freedom.
Problems[edit]
The Nosé-Hoover thermostat has problems with ergodicity for small or stiff systems. In order to compensate for this a modification using "chains" has been proposed [4].
Non-equilibrium[edit]
A version of the Nosé-Hoover thermostat has been developed for non-equilibrium simulations [5].
References[edit]
- ↑ Shuichi Nosé "A unified formulation of the constant temperature molecular dynamics methods" , Journal of Chemical Physics 81 pp. 511-519 (1984)
- ↑ Shuichi Nosé "A molecular dynamics method for simulations in the canonical ensemble", Molecular Physics 52 pp. 255-268 (1984)
- ↑ William G. Hoover "Canonical dynamics: Equilibrium phase-space distributions", Physical Review A 31 pp. 1695-1697 (1985)
- ↑ Glenn J. Martyna, Michael L. Klein and Mark Tuckerman "Nosé–Hoover chains: The canonical ensemble via continuous dynamics", Journal of Chemical Physics 97 pp. 2635- (1992)
- ↑ Ben Leimkuhler, Frédéric Legoll and Emad Noorizadeh "A temperature control technique for nonequilibrium molecular simulation", Journal of Chemical Physics 128 074105 (2008)
Related reading
- D. J. Evans and B. L. Holian "The Nose–Hoover thermostat", Journal of Chemical Physics 83 pp. 4069-4074 (1985)
- Carlos Braga and Karl P. Travis "A configurational temperature Nosé-Hoover thermostat", Journal of Chemical Physics 123 134101 (2005)
- See http://williamhoover.info and Wm. G. Hoover and Carol G. Hoover "Time Reversibility, Computer Simulations, Algorithms, Chaos", Advanced Series in Nonlinear Dynamics 13 World Scientific (2012) ISBN 978-981-4383-16-5