Ergodicity of Thermostat Family of Nos\'e--Hoover type

摘要

One-variable thermostats are studied as a generalization of theNos\'e--Hoover method which is aimed to achieve Gibbs' canonical distributionwith conserving the time-reversibility. A condition for equations of motion forthe system with the thermostats is derived in the form of a partialdifferential equation. Solutions of this equation construct a family ofthermostats including the Nos\'e--Hoover method as the minimal solution. It isshown that the one-variable thermostat coupled with the one-dimensionalharmonic oscillator loses its ergodicity with large enough relaxation time. Thepresent result suggests that multi-variable thermostats are required to assurethe ergodicity and to work as heatbath.