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Non-Ergodicity of Nose-Hoover chain thermostat in computationally achievable time

The widely used Nose-Hoover chain (NHC) thermostat in molecular dynamics simulations is generally believed to impart the canonical distribution as well as quasi- (i.e., space filling) ergodicity on the thermostatted physical system (PS). Working with the standard single harmonic oscillator, we prove analytically that the two chain Nose-Hoover thermostat with unequal thermostat masses approach the standard Nose-Hoover dynamics and hence the PS loses its canonical and quasi-ergodic nature. We also show through numerical simulations over substantially long times that for certain Poincare sections, for both the equal and unequal thermostat mass cases, the bivariate distribution function of position and momentum (${x,p}$) and of reservoir degrees of freedom ($ξ,η$) lose their Gaussian nature. Further, the 4-dimensional $x-p-ξ-η$ extended phase space exhibits two holes of non-zero measure. The NHC thermostat therefore does not generate the canonical distribution or preserve quasi-ergodicity for the PS.