# Nosé–Hoover thermostat

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The Nosé–Hoover thermostat is a deterministic method used in molecular dynamics to keep the temperature around an average. It was originally introduced by Nosé and developed further by Hoover. The heat bath is made into an integral part of the system by adding an artificial variable associated with an artificial mass.

## Introduction

In classic molecular dynamics, simulations are done in the microcanonical ensemble, meaning that we control the volume, the number of particles and the energy. In real life however, we control the temperature instead of the energy. Because of the nature of the simulation, it's not possible to switch from the microcanonical ensemble to the canonical ensemble in which we control the temperature instead of the energy. Several methods have been introduced to keep the temperature constant while using the microcanonical ensemble. Popular techniques to control temperature include velocity rescaling, the Andersen thermostat, the Nosé–Hoover thermostat, Nosé–Hoover chains, the Berendsen thermostat and Langevin dynamics.

The central idea is to simulate in such a way that we obtain a canonical distribution: this means fixing the average temperature of the system under simulation, but at the same time allowing for a fluctuation of the temperature with a distribution typical for a canonical distribution.

## The Nosé-Hoover thermostat

In the approach of Nosé, we introduce a heatbath in the Hamiltonian with an extra degree of freedom s. The total Hamiltonian we will use to simulate is,

$\mathcal{H} (P,R,p_s,s) = \sum_i \frac{\mathbf{p}_i^2}{2ms^2} + \frac12 \sum_{ij,i\not= j} U \left( \mathbf{r_i} - \mathbf{r_j}\right) + \frac{p_s^2}{2Q} + gkT\ln\left( s\right),$

where g is the number of independent momentum degrees of freedom of the system, R and P represent all coordinates $\mathbf{r_i}$ and $\mathbf{p_i}$ and Q is a parameter that should be chosen carefully. The coordinates R, P and t in this Hamiltonian are virtual. They are related to the real coordinates as follows:

$R'=R,~ P'=\frac{P}{s} ~\text{and}~t'=\int^t \frac{\mathrm{d}\tau}{s}$,

where the coordinates with an accent are the real coordinates. It can be shown that when using this Hamiltonian, taking a microcanonical ensemble average is the same as a canonical ensemble average when $g=3N$.

## References

• Nose, S (1984). "A unified formulation of the constant temperature molecular-dynamics methods". Journal of chemical physics 81 (1): 511–519. doi:10.1063/1.447334.
• Hoover, William G. (Mar 1985). "Canonical dynamics: Equilibrium phase-space distributions". Phys. Rev. A (American Physical Society) 31 (3): 1695–1697. doi:10.1103/PhysRevA.31.1695.
• Thijssen, J. M. (2007). Computational Physics (2nd ed.). Cambridge University Press. pp. 226–231. ISBN 978-0-521-83346-2.