Markovian Approximation for the Nosé–Hoover Method and H-Theorem

Abstract

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.