William G. Hoover

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.

In order to confirm the existence of a first-order melting transition for a classical many-body system of hard spheres and to discover the densities of the coexisting phases, we have made a Monte Carlo determination of the pressure and absolute entropy of the hard-sphere solid. We use these solid-phase thermodynamic properties, coupled with known fluid-phase data, to show that the hard-sphere solid, at a density of 0.74 relative to close packing, and the hard-sphere fluid, at a density of 0.67 relative to close packing, satisfy the thermodynamic equilibrium conditions of equal pressure and chemical potential at constant temperature. To get the solid-phase entropy, we integrated the Monte Carlo pressure–volume equation of state for a “single-occupancy” system in which the center of each hard sphere was constrained to occupy its own private cell. Such a system is no different from the ordinary solid at high density, but at low density its entropy and pressure are both lower. The difference in entropy between an unconstrained system of particles and a constrained one, with one particle per cell, is the so-called “communal entropy,” the determination of which has been a fundamental problem in the theory of liquids. Our Monte Carlo measurements show that communal entropy is nearly a linear function of density.

Some of the differences among several alternative formulations of constant-pressure molecular dynamics are described. The formulations all agree in the large-system limit, but differ for small systems.

Recent experiments at strain rates reaching 0.1 GHz suggest a power-law dependence of solid-phase shear stress on strain rate. Novel nonequilibrium molecular dynamics simulations of plastic flow have been carried out. These steady-state isothermal calculations appear to be consistent with the present-day experimental data and suggest that the flows of metals can be described by a single physical mechanism over a range of strain rates from 10 kHz to 1 THz.