Nosé–Hoover thermostat

The Nosé–Hoover thermostat is a deterministic algorithm for constant-temperature molecular dynamics simulations. It was originally developed by Nosé and was improved further by Hoover. Although the heat bath of Nosé–Hoover thermostat consists of only one imaginary particle, simulation systems achieve realistic constant-temperature condition (canonical ensemble). Therefore, the Nosé–Hoover thermostat has been commonly used as one of the most accurate and efficient methods for constant-temperature molecular dynamics simulations.

Introduction

In classic molecular dynamics, simulations are done in the microcanonical ensemble; a number of particles, volume, and energy have a constant value. In experiments, however, the temperature is generally controlled instead of the energy. The ensemble of this experimental condition is called a canonical ensemble. Importantly, the canonical ensemble is perfectly different from microcanonical ensemble from the viewpoint of statistical mechanics. 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, a Hamiltonian with an extra degree of freedom for heat bath, s, is introduced;

${\mathcal {H}}(P,R,p_{s},s)=\sum _{i}{\frac {\mathbf {p} _{i}^{2}}{2ms^{2}}}+{\frac {1}{2}}\sum _{ij,i\not =j}U\left(\mathbf {r_{i}} -\mathbf {r_{j}} \right)+{\frac {p_{s}^{2}}{2Q}}+gkT\ln \left(s\right),$

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 an imaginary mass which should be chosen carefully along with systems. 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 should be noted that the ensemble average of the above Hamiltonian at $g=3N$ is equal to the canonical ensemble average.

Hoover (1985) used the phase-space continuity equation, a generalized Liouville equation, to establish what is now known as the Nosé–Hoover thermostat. This approach does not require the scaling of the time (or, in effect, of the momentum) by s.

References

Nosé, S (1984). "A unified formulation of the constant temperature molecular-dynamics methods". Journal of Chemical Physics. 81 (1): 511–519. Bibcode:1984JChPh..81..511N. doi:10.1063/1.447334.

Hoover, William G. (Mar 1985). "Canonical dynamics: Equilibrium phase-space distributions". Phys. Rev. A. 31 (3). American Physical Society: 1695–1697. Bibcode:1985PhRvA..31.1695H. 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.

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