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

TL;DR

This paper derives a generalized Einstein relation for a Langevin equation with state-dependent noise and demonstrates that the Nosé--Hoover dynamics, under a Markovian approximation, satisfies the H-theorem, functioning effectively as a heat bath.

Contribution

It introduces a Markovian approximation for the Nosé--Hoover method and proves the validity of the H-theorem for its coarse-grained dynamics.

Findings

Generalized Einstein relation holds for the associated Fokker--Planck equation.

Nosé--Hoover dynamics satisfy the H-theorem under the approximation.

The method confirms the Nosé--Hoover dynamics as a heat bath.

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 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.