Hard Sphere Dynamics for Normal and Granular Fluids

A fluid of N smooth, hard spheres is considered as a model for normal (elastic collisions) and granular (inelastic collisions) fluids. The potential energy is discontinuous for hard spheres so the pairwise forces are singular and the usual forms of Newtonian and Hamiltonian mechanics do not apply. Nevertheless, particle trajectories in the N particle phase space are well defined and the generators for these trajectories can be identified. The first part of this presentation is a review of the generators for the dynamics of observables and probability densities. The new results presented in the second part refer to applications of these generators to the Liouville dynamics for granular fluids. A set of eigenvalues and eigenfunctions of the generator for this Liouville dynamics is identified in a special "stationary representation". This provides a class of exact solutions to the Liouville equation that are closely related to hydrodynamics for granular fluids.Comment: Submitted for publication in the Proceedings of Workshop on Nonlinear Dynamics in Astronomy and Physics, eds. S. Gottesmann and J. R. Buchler (Annals of the New York Academy of Sciences, 2005

Shear Stress Correlations in Hard and Soft Sphere Fluids

The shear stress autocorrelation function has been studied recently by molecular dynamics simulation using the 1/q^n potential for very large n. The results are analyzed and interpreted here by comparing them to the shear stress response function for hard spheres. It is shown that the hard sphere response function has a singular contribution and that this is reproduced accurately by the simulations for large n. A simple model for the stress autocorrelation function at finite n is proposed, based on the required hard sphere limiting form.Comment: 14 pages, 2 figures; submitted for special issue of Molecular Physic

Nonequilibrium Phase Transition for a Heavy Particle in a Granular Fluid

It is shown that the homogeneous cooling state (HCS) for a heavy impurity particle in a granular fluid supports two distinct phases. The order parameter $\phi$ is the mean square velocity of the impurity particle relative to that of a fluid particle, and the control parameter $\xi$ is the fluid cooling rate relative to the impurity collision rate. For $\xi<1$ there is a ``normal'' phase for which $\phi$ scales as the fluid/impurity mass ratio, just as for a system with elastic collisions. For $\xi>1$ an ``ordered'' phase occurs in which $\phi$ is finite even for vanishingly small mass ratio, representing an extreme violation of energy equipartition. The phenomenon can be described in terms of a Landau-like free energy for a second order phase transition. The dynamics leading to the HCS is studied in detail using an asymptotic analysis of the Enskog-Lorentz kinetic equation near each phase and the critical domain. Critical slowing is observed with a divergent relaxation time at the critical point. The stationary velocity distributions are determined in each case, showing a crossover from Maxwellian in the normal phase to an exponential quartic function of the velocity that is sharply peaked about the non-zero $\phi$ for the ordered phase. It is shown that the diffusion coefficient in the normal phase diverges at the critical point and remains so in the ordered phase. This is interpreted as a transition from diffusive to ballistic dynamics between the normal and ordered phases.Comment: 28 pages, 12 figures include