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 chain. Although the underlining dynamics is non-Hamiltonian, the corresponding equations of motion conserve the following energy term:

$\mathcal{H'} = \mathcal{H}(\bold{r},\bold{p}) + \sum\limits_{j=1}^{M} \frac{p_{\eta_j}^2}{2Q_j} + dNk_{B} T \theta_1 + k_{B} T \sum\limits_{j=2}^{M} \theta_j$