# 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)

where is the thermodynamic friction coefficient, given by (Ref. 3 Eq. 5)

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