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(upsilon) and v similar to 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.