William G. Hoover

Melting Transition and Communal Entropy for Hard Spheres

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 betwee...

Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks

New expressions for the fourth, fifth, and sixth virial coefficients are obtained as sums of modified star integrals. The modified stars contain both Mayer f functions and f functions (f≡f+1). It is shown that the number of topologically distinguishable graphs occurring in the new expressions is about half the number required in previous expressions. This reduction in the number of integrals makes numerical calculation of virial coefficients simpler and more nearly accurate. For particles interacting with a hard‐core potential, values of the modified star integrals are shown to depend strongly on dimension; for example, several modified star integrals are identically zero for hard disks (two dimensions), but give nonzero values for hard spheres (three dimensions). Of all the modified star integrals contributing to the fourth, fifth, and sixth virial coefficients, the complete star integrals are shown to be the largest. Mayers expressions for these coefficients made the complete star integrals the small...

Studies in Molecular Dynamics. V. High‐Density Equation of State and Entropy for Hard Disks and Spheres

The equations of state for periodic systems of hard disks and hard spheres in the solid phase have been accurately determined and used to evaluate the coefficients in the expansion of the pressure in powers of the relative free volume, α = (V − V0) / V0, where V0 is the close‐packed volume. For disks pV / NkT = 2 / α + 1.90 + 0.67α + O(α2) and for spheres pV / NkT = 3 / α + 2.56 + 0.56α + O(α2). These coefficients are compared to cell models, and those models which include correlations between neighboring particles work best. An equivalent expansion of other thermodynamic properties requires the entropy constant to be evaluated in the close‐packed limit. This constant is obtained here by integrating the equation of state over the entire density region. The Lennard‐Jones–Devonshire cell‐theory estimates of the entropy constant are nearly correct; that is, the cell‐theory estimate is too small by 0.06Nk for disks and too large by 0.24Nk for spheres. The pressure difference and hence the entropy difference b...

Thermodynamic Properties of the Fluid and Solid Phases for Inverse Power Potentials

The two computer methods of Monte Carlo and lattice dynamics are used to determine fluid and face‐centered‐cubic solid thermodynamic properties for classical particles interacting with pairwise‐additive inverse 4th, 6th, and 9th power potentials. These results, together with those already on hand for 12th power and hard‐sphere potentials, provide a complete, and remarkably simple, description of the dependence of the pure‐phase thermodynamics and the melting transition on the “softness” of the pair potential.

Soft‐Sphere Equation of State

The pressure and entropy for soft‐sphere particles interacting with an inverse twelfth‐power potential are determined using the Monte Carlo method. The solid‐phase entropy is calculated in two ways: by integrating the single‐occupancy equation of state from the low density limit to solid densities, and by using solid‐phase Monte Carlo pressures to evaluate the anharmonic corrections to the lattice‐dynamics high‐density limit. The two methods agree, and the entropy is used to locate the melting transition. The computed results are compared with the predictions of the virial series, lattice dynamics, perturbation theories, and cell models. For the fluid phase, perturbation theory is very accurate up to two‐thirds of the freezing density. For the solid phase, a correlated cell model predicts pressures very close to the Monte Carlo results.

Seventh Virial Coefficients for Hard Spheres and Hard Disks

The seventh virial coefficient B7 is expressed as a sum of modified star integrals instead of the usual Mayer star integrals. The new graphs contain both Mayer f functions and f(≡1+f) functions. This simplifies the calculation of B7. That is, instead of the 468 star integrals that appear in Mayers formulation, only 171 modified star integrals now appear in the evaluation of B7. Furthermore, for D‐dimensional particles with a hard core, these 171 integrals are strongly dependent on the number of dimensions D. For hard rods, hard disks, and hard spheres, respectively, at most one, 78, and 164 of these integrals give a nonvanishing contribution to B7. When Monte Carlo integration is used to evaluate these integrals using hard‐sphere and hard‐disk potentials we obtain the following values of B7: hard spheres, B7/(B2)6=0.0138±0.0004; hard disks, B7/(B2)6=0.1141±0.0005. For hard spheres, the truncated seven‐term virial series for the pressure agrees within 10% with the results of the molecular dynamics data t...

Diffusion in a periodic Lorentz gas

We use a constant “driving force”Fd together with a Gaussian thermostatting “constraint force”Fd to simulate a nonequilibrium steady-state current (particle velocity) in a periodic, two-dimensional, classical Lorentz gas. The ratio of the average particle velocity to the driving force (field strength) is the Lorentz-gas conductivity. A regular “Galton-board” lattice of fixed particles is arranged in a dense triangular-lattice structure. The moving scatterer particle travels through the lattice at constant kinetic energy, making elastic hard-disk collisions with the fixed particles. At low field strengths the nonequilibrium conductivity is statistically indistinguishable from the equilibrium Green-Kubo estimate of Machta and Zwanzig. The low-field conductivity varies smoothly, but in a complicated way, with field strength. For moderate fields the conductivity generally decreases nearly linearly with field, but is nearly discontinuous at certain values where interesting stable cycles of collisions occur. As the field is increased, the phase-space probability density drops in apparent fractal dimensionality from 3 to 1. We compare the nonlinear conductivity with similar zero-density results from the two-particle Boltzmann equation. We also tabulate the variation of the kinetic pressure as a function of the field strength,

Studies in Molecular Dynamics. IV. The Pressure, Collision Rate, and Their Number Dependence for Hard Disks

The pressure for four, 12, and 72 hard disks determined dynamically from the virial theorem or the collision rate is shown to be identical to that determined by the Monte Carlo method. To show this equivalence, it is necessary to take into account that the center‐of‐mass velocity is kept fixed in the dynamic system. This numerical agreement suggests the validity of the quasiergodic hypothesis even for small systems. The (lnN)/N dependence of the phase‐transition pressure on the number of particles N is simply explained in terms of the communal entropy.

Statistical Mechanics of Phase Diagrams. I. Inverse Power Potentials and the Close‐Packed to Body‐Centered Cubic Transition

Most of the softer metals exhibit a low‐temperature close‐packed phase, an intermediate‐temperature body‐centered phase, and a high‐temperature fluid phase. Here we relate this behavior to that of theoretical model systems in which particles interact with the inverse power potential φ (r)=e ( σ / r)n. We show that the same three‐phase behavior occurs for the models provided that the interparticle repulsion is sufficiently soft (n ≤ 7). For the model systems the phase boundary between the close‐packed and the body‐centered phases is located using lattice dynamics. The fluid—solid melting line is deduced from Monte Carlo computer experiments.

Exact hard‐disk free volumes

Properties of exact hard‐disk free volumes are determined by a combination of analytical and numerical techniques. Both the high‐density fluid phase and the lower‐density fluid phase are treated. These one‐particle free volumes are used to verify known thermodynamic information for hard disks and to calculate the shear modulus for the hard‐disk solid phase. The free volumes are also compared to approximate free‐volume estimates made from the known hard‐disk entropy. The statistical distributions of free volume and free surface (perimeter of the free volume) are studied. The percolation transition, at which the free‐volume changes from extensive to intensive, is found to occur at about one‐third of the freezing density.