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A Langevin equation with state-dependent random force is considered. When theHelmholtz free energy is a nonincreasing function of time (the H-theorem), ageneralized Einstein relation is obtained. A stochastic process of theNos\'e--Hoover method is discussed on the basis of the Markovian approximation.It is found that the generalized Einstein relation holds for the Fokker--Planckequation associated with the stochastic Nos\'e--Hoover equation. The presentresult indicates that the Nos\'e--Hoover dynamics coarse-grained with timesatisfies the H-theorem and therefore works as a heat bath.

Markovian Approximation for the Nos\'e--Hoover method and H-theorem