Nosé-Hoover thermostat
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)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\mathrm {d}}{\mathbf{v}}(t)}{{\mathrm {d}t}} = \frac{{\mathbf {F}}(t)}{m} -\zeta {\mathbf{v}}(t)}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta} is the thermodynamic friction coefficient, given by (Ref. 3 Eq. 5)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\mathrm {d}}\zeta(t)}{{\mathrm {d}t}} = \frac{1}{Q} \left[ \sum m {\mathbf{v}}(t)^2 - (X+1)k_BT \right]}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is a parameter that has the dimensions of energyFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times} (time)2 and determines the time-scale of the temperature fluctuation and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} 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