Nose-Hoover-chain thermostat

The standard Nose Hoover suffers from well known issues, such as the ergodicity violation in the case of simple harmonic oscillator. As proposed by Martyna and Klein, these problems can be solved by using multiple Nose Hoover thermostats connected in a chain. Although the underlining dynamics is non-Hamiltonian, the corresponding equations of motion conserve the following energy term:

$\mathcal{H'} = \mathcal{H}(\bold{r},\bold{p}) + \sum\limits_{j=1}^{M} \frac{p_{\eta_j}^2}{2Q_j} + dNk_{B} T \eta_1 + k_{B} T \sum\limits_{j=2}^{M} \eta_j$

where $\mathcal{H}(\bold{r},\bold{p})$ is the Hamiltonian of the physical system, $M$ is the number of thermostats, and $\eta_{j}$, $p_{\eta_j}$, and $Q_{j}$ are the position, momentum, and mass-like parameter associated with the thermostat $j$.