Particle dynamics in hard-sphere systems

The Fourier spectra of the number density, convection and force fields are obtained numerically for an assembly of hard-spheres. The magnitude spectra determine the dominant wave numbers, and the phase difference between the hard-sphere force and number density spectra determines the nature of particle dynamics. The latter is used to show that for every wave number k there is a critical frequency ω-c$/(k), such that when ω>ω-c$/(k) the phase difference is -π/2 and when ω<ω-c$/(k) the phase difference is π/2. This change in the phase difference corresponds to a change of type of the governing partial differential equations from hyperbolic (for modes with ω>ω-c$/(k)) to elliptic (for modes with ω<ω-c$/(k)). The convection forces are shown to be of the same order of magnitude as the hard-sphere forces. The change of type of the equations from hyperbolic to elliptic is possible because the magnitude of the convection term is comparable to the magnitude of the force term. These results are similar to the results obtained previously for the Lennard-Jones liquids. One significant difference, however, is that the dominant off k-axis peaks of the Lennard-Jones force spectra are replaced by the bands that are at approximately the same dimensionless wave numbers.