Single-Speed Molecular Dynamics of Hard Parallel Squares and Cubes

TL;DR

This paper introduces a novel single-speed molecular dynamics method to study the equations of state for hard squares and cubes, confirming that configurational properties do not depend on velocity distribution and exploring phase transition behaviors.

Contribution

It presents a new single-speed molecular dynamics approach and applies it to hard squares and cubes, providing insights into their phase transitions and validating configurational independence from velocity distribution.

Findings

Excellent agreement with previous models and simulations.

Hard squares show a second-order melting transition at density 0.79.

Cubes exhibit complex behavior with no clear first-order transition.

Abstract

The fluid and solid equations of state for hard parallel squares and cubes are reinvestigated here over a wide range of densities. We use a novel single-speed version of molecular dynamics. Our results are compared with those from earlier simulations, as well as with the predictions of the virial series, the cell model, and Kirkwood's many-body single-occupancy model. The single-occupancy model is applied to give the absolute entropy of the solid phases just as was done earlier for hard disks and hard spheres. The excellent agreement found here with all relevant previous work shows very clearly that configurational properties, such as the equation of state, do not require the maximum-entropy Maxwell-Boltzmann velocity distribution. For both hard squares and hard cubes the free-volume theory provides a good description of the high-density solid-phase pressure. Hard parallel squares appear to exhibit a second-order melting transition at a density of 0.79 relative to close-packing. Hard parallel cubes have a more complicated equation of state, with several relatively-gentle curvature changes, but nothing so abrupt as to indicate a first-order melting transition. Because the number-dependence for the cubes is relatively large the exact nature of the cube transition remains unknown.