Inelastic collapse and clumping in a one-dimensional granular medium

The dynamics of a one‐dimensional gas of inelastic point particles is investigated. To model inelastic collisions, it is supposed that the relative velocity of two colliding particles is reduced by a factor r, where 0<r<1. The constant r is the coefficient of restitution. Because the collisions are inelastic, particles can collide infinitely often in finite time so that the relative separations and velocities of adjacent particles on the line become zero. The minimal example of this ‘‘inelastic collapse’’ requires r<7−4 7/8 ≊0.0718. With this restriction, three particles condense into a single lump in a finite time: The particle in the middle is sandwiched between the monotonically converging outer particles. When r is greater than 7−4 7/8, more than three particles are needed to trigger inelastic collapse and it is shown that r is close to 1 the minimum number scales as −ln(1−r)/(1−r). The simplest statistical problem is the ‘‘cooling law’’ of a uniformly excited gas confined between inelastic boundaries...