Nose-Hoover thermostat
In the approach by Nosé and Hoover[1][2][3] 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 [math]\displaystyle{ s }[/math] which is introduced into the Lagrangian of the system. This Lagrangian for an [math]\displaystyle{ N }[/math] is written as
[math]\displaystyle{ P(t)=\nu e^{-\nu t}. }[/math]
References