7.1.2 Nosé-Hoover dynamics

Instead of perfectly constraining the kinetic energy, one may introduce a sort of thermostat which acts to keep the energy near a desired value. Nosé has shown that under quite general conditions the spread around the given $E_{0}$ is such that a canonical distribution is approached.

Hoover rewrote Nosé's original equations of motion in the following simple way:

$$\dot{\vec{r}} = \frac{\vec{p}}{m}$$ (7.7)

$$\dot{\vec{p}} = \vec{K}_{i} - \zeta \vec{p}$$ (7.8)

$$\dot{\zeta} = \frac{2}{Q} \left[ E_{kin} - E_{0} \right]$$ (7.9)

where we have omitted the particle index $i$. The coupling parameter $Q^{-1}$ represents the efficiency of the thermal bath holding the temperature constant. $\zeta$ is a kind of ``viscosity'' which, however, may take on negative values.

Nosé has shown that the phase space trajectory of such a system will under very general assumptions (a few interacting particles suffice) lead to canonical averaging. (See also Posch, Hoover, Vesely Phys.Rev.A 33(1986)4253.)

For $Q \rightarrow 0$ the spread of $E_{kin}$ around $E_{0}$ approaches zero, and the Nosé-Hoover dynamics becomes identical to Gauss dynamics.