Generalization of Nose and Nose-hoover isothermal dynamics

The infinitely many possible isothermal dynamics based on Nose and Nose-Hoover methods are investigated. Their properties and criteria for selecting different isothermal dynamics determined by various scaling functions of the thermostat s variable involved in the generalized Nose Hamiltonian [J. Jellinek and R. S. Berry, Phys. Rev. A 38, 3069 (1988)] are tested with molecular dynamics simulations, and examined analytically. It is shown that time scaling is related to the scaling of the momenta. It is demonstrated that, for practical realizations, the entire generalization of the Nose-Hoover method reduces to only two momentum scaling functions h and u, with a function v defining the "potential energy" of the thermostat. The most general form of the generalized Nose-Hoover (GNH) equations of motion is established. It enables correct calculations of both static and dynamic equilibrium quantities. GNH equations with h=s(alpha), u=s(vartheta), and v approximately lns are studied in detail. With such a choice of the functions the extended Nose-Hoover (ENH) equations are expected to produce more chaotic phase-space dynamics than the NH equations. This is illustrated by thermalization of a one dimensional harmonic oscillator. For a system away from equilibrium the ENH thermostat is not able to provide dynamics consistent with the target temperature, and, thus, the GNH approach reduces to the original Nose-Hoover thermostat. A simple modification of the ENH equations is proposed which makes the ENH thermostat also applicable to nonequilibrium states.