Canonical dynamics of the Nosé oscillator: Stability, order, and chaos.

Abstract

Nos\'e has developed many-body equations of motion designed to reproduce Gibbs's canonical phase-space distribution. These equations of motion have a Hamiltonian basis and are accordingly time reversible and deterministic. They include thermodynamic temperature control through a single deterministic friction coefficient, which can be thought of as a control variable or as a memory function. We apply Nos\'e's ideas to a single classical one-dimensional harmonic oscillator. This relatively simple system exhibits both regular and chaotic dynamical trajectories, depending on the initial conditions. We explore here the nature of these solutions by estimating their fractal dimensionality and Lyapunov instability. The Nos\'e oscillator is a borderline case, not sufficiently chaotic for a fully statistical description. We suggest that the behavior of only slightly more complicated systems is considerably simpler and in accord with statistical mechanics.