KINETICS OF A ONE-DIMENSIONAL GRANULAR MEDIUM IN THE QUASI-ELASTIC LIMIT

The dynamics of a one-dimensional granular medium has a finite time singularity if the number of particles in the medium is greater than a certain critical value. The singularity (''inelastic collapse'') occurs when a group of particles collides infinitely often in a finite time so that the separations and relative velocities vanish. To avoid the finite time singularity, a double limit in which the coefficient of restitution r approaches 1 and the number of particles N becomes large, but is always below, the critical number needed to trigger collapse, is considered. Specifically, r --> 1 with N approximately (1 - r)-1. This procedure is called the ''quasielastic'' limit. Using a combination of direct simulation and kinetic theory, it is shown that a bimodal velocity distribution develops from random initial conditions. The bimodal distribution is the basis for a ''two-stream'' continuum model in which each stream represents one of the velocity modes. This two-stream model qualitatively explains some of the unusual phenomena seen in the simulations, such as the growth of large-scale instabilities in a medium that is excited with statistically homogeneous initial conditions. These instabilities can be either direct or oscillatory, depending on the domain size, and their finite-amplitude development results in the formation of clusters of particles.