Ergodicity of a thermostat family of the Nosé-Hoover type

One-variable thermostats are studied as a generalization of the Nosé-Hoover method, which is aimed at achieving Gibbs’ canonical distribution while conserving the time reversibility. A condition for equations of motion for the system with the thermostats is derived in the form of a partial differential equation. Solutions of this equation constitute a family of thermostats including the Nosé-Hoover method as the minimal solution. It is shown that the one-variable thermostat coupled with the one-dimensional harmonic oscillator loses its ergodicity with large enough relaxation time. The present result suggests that multivariable thermostats are required to assure the ergodicity and to work as a heat bath.

Figure 1

The range of the energy $H_0$. The function $\varphi \left(H_0\right)=H_0-(m+1){\beta }^{-1}\ln H_0$ and $C$ are plotted for $m=0$, $\beta =1$, and $C=1.019$. The minimum and the maximum values are determined as solutions of $\varphi \left(H_0\right)=C$. Note that this equation always has two positive solutions $H_{\min }$ and $H_{\max }$. From the inequality (15), we can estimate the range of the energy as $0.815⩽H_0⩽1.211$.

Figure 2

(Color online) Time evolutions of the energies of the systems with three thermostats (a) $g=p\zeta$, (b) $g=p{\zeta }^3$, and (c) $g=p^3\zeta$. Three cases $\tau =5$, 10, and 50 are plotted in each figure. The upper and the lower limits of energies estimated from the inequality (15) are shown as the solid lines. The theoretical estimation becomes more accurate with larger $\tau$.