By Florian Cajori
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In Eq. 14), the ﬂuctuating force ????th = (1∕Nth ) i=1 Si ????i , which ensures that the macroscopic momentum of the thermostatted particles is a constant of the motion, which we set to zero. The Nosé–Hoover thermostat adds a degree of freedom to the phase space of the system, and the extended phase space vector is ????∗ ≡ (????, ????). However, in order to simplify the notation, from here on we represent this implicitly using ????. Deﬁnition If in the absence of the thermostatting terms, the equations of motion preserve the phase space volume, Λad ≡ (????∕????????) ⋅ ????̇ ad = 0, and the system is said to satisfy the condition known as the adiabatic incompressibility of phase space or AI????, (Evans and Morriss, 1990).