Canonical dynamics: Equilibrium phase-space distributions

Nos\'e has modified Newtonian dynamics so as to reproduce both the canonical and the isothermal-isobaric probability densities in the phase space of an N-body system. He did this by scaling time (with s) and distance (with ${V}^{1/D}$ in D dimensions) through Lagrangian equations of motion. The dynamical equations describe the evolution of these two scaling variables and their two conjugate momenta ${p}_{s}$ and ${p}_{v}$. Here we develop a slightly different set of equations, free of time scaling. We find the dynamical steady-state probability density in an extended phase space with variables x, ${p}_{x}$, V, \ensuremath{\epsilon}\ifmmode \dot{}\else \.{}\fi{}, and \ensuremath{\zeta}, where the x are reduced distances and the two variables \ensuremath{\epsilon}\ifmmode \dot{}\else \.{}\fi{} and \ensuremath{\zeta} act as thermodynamic friction coefficients. We find that these friction coefficients have Gaussian distributions. From the distributions the extent of small-system non-Newtonian behavior can be estimated. We illustrate the dynamical equations by considering their application to the simplest possible case, a one-dimensional classical harmonic oscillator.