Kinetics of a one‐dimensional granular medium in the quasielastic 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∼(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 s...