Nonequilibrium thermodynamics and nose-hoover dynamics

We show that systems driven by an external force and described δ_iS = K_bD[f(t, γ(t)) 7∥ f_s (t, γ_s (t)) f_r ^eq (ζ(t)] ≥ 0 by Nose-Hoover dynamics allow for a consistent nonequilibrium thermodynamics description when the thermostatted variable is initially assumed in a state of canonical equilibrium. By treating the "real" variables as the system and the thermostatted variable as the reservoir, we establish the first and second law of thermodynamics. As for Hamiltonian systems, the entropy production can be expressed as a relative entropy measuring the system-reservoir correlations established during the dynamics.