Markovian approximation for the Nosé-Hoover method and H-theorem

A Langevin equation with state-dependent random force is considered. When the Helmholtz free energy is a nonincreasing function of time (the H-theorem), a generalized Einstein relation is obtained. A stochastic process of the Nosé-Hoover method is discussed on the basis of the Markovian approximation. It is found that the generalized Einstein relation holds for the Fokker-Planck equation associated with the stochastic Nosé-Hoover equation. The present result indicates that the Nosé-Hoover dynamics coarse-grained with time satisfies the H-theorem and therefore the canonical distribution is guaranteed for the subsystem.