Nose-Hoover thermostat

In the approach by Nosé and Hoover an extra degree of freedom is introduced in the Hamiltonian. The heat bath is considered as an integral part of the system and has a fictious coordinate $s$ which is introduced into the Lagrangian of the system. This Lagrangian for an $N$ is written as

${\mathcal {L}}=\sum \limits _{i=1}^{N}{\frac {m_{i}}{2}}s^{2}{\dot {\mathbf {r}}}_{i}^{2}-U({\mathbf {r}})+{\frac {Q}{2}}{\dot {s}}^{2}-{\frac {g}{\beta }}\mathrm {ln} s.$

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