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 Nos\'e--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 Nos\'e--Hoover equation. The present result indicates that the Nos\'e--Hoover dynamics coarse-grained with time satisfies the H-theorem and therefore works as a heat bath.