Ergodicity of Isothermal Molecular Dynamics Method

A condition for equations of motion for isothermal dynamics is derived, and the Nos\'e--Hoover method is generalized on the basis of this condition. The ergodicity of the one-variable thermostats are studied, and it is shown that the one-variable thermostat coupled with the one-dimensional harmonic oscillator loses its ergodicity with large enough relaxation time. A stochastic process of the Nos\'e--Hoover method is also discussed based on the Markovian approximation.